ABELIAN TENSORS
J.M. LANDSBERG AND MATEUSZ MICHALEK
Abstract. We analyze tensors in Cm ⊗Cm ⊗Cm satisfying Strassen’s equations for border rank
m. Results include: two purely geometric characterizations of the Coppersmith-Winograd tensor, a reduction to the study of symmetric tensors under a mild genericity hypothesis, and
numerous additional equations and examples. This study is closely connected to the study of
the variety of m-dimensional abelian subspaces of End(Cm ) and the subvariety consisting of
the Zariski closure of the variety of maximal tori, called the variety of reductions.
1. Introduction
The rank and border rank of a tensor T ∈ Cm ⊗Cm ⊗Cm (defined below) are basic measures
of its complexity. Central problems are to develop techniques to determine them (see, e.g.,
[27, 12, 14, 22]). Complete resolutions of these problems are currently out of reach. For example,
neither problem is solved already in C4 ⊗C4 ⊗C4 . This article focuses on a very special class of
tensors, those satisfying Strassen’s commutativity equations (see §2.1). The study of such tensors
is related to the classical problem of studying spaces of commuting matrices, see, e.g. [20, 42,
21, 26].
To completely understand border rank, it would be sufficient to understand the case of border
rank m in Cm ⊗Cm ⊗Cm . We study this problem under two genericity hypotheses - concision,
which essentially says we restrict to tensors that are not contained in some Cm−1 ⊗Cm ⊗Cm , and
1A -genericity, which is defined below. Even under these genericity hypotheses, the problem is
still subtle.
Let A, B, C be complex vector spaces of dimensions a, b, c, let T ∈ A⊗B⊗C be a tensor.
(In bases T is a three dimensional matrix of size a × b × c.) We may view T as a linear map
T ∶ A∗ → B⊗C ≃ Hom(C ∗ , B). (In bases, T ((α1 , ⋯, αa )) is the b × c matrix α1 times the first
slice of the a × b × c matrix, plus α2 times the second slice ... plus αa times the a-th slice.) One
may recover T up to isomorphism from the space of linear maps T (A∗ ).
One says T has rank one if T = a⊗b⊗c for some a ∈ A, b ∈ B and c ∈ C, and the rank of
T , denoted R(T ) is the smallest r such that T may be expressed as the sum of r rank one
tensors. Rank is not semi-continuous, so one defines the border rank of T , denoted R(T ), to
be the smallest r such that T is a limit of tensors of rank r, or equivalently (see e.g. [27, Cor.
5.1.1.5]) the smallest r such that T lies in the Zariski closure of the set of tensors of rank r.
Write σ̂r (Seg(PA × PB × PC)) ⊂ A⊗B⊗C for the variety of tensors of border rank at most r
(the cone over the r-th secant variety of the Segre variety). We will be mostly concerned with
the case a = b = c = m.
To make the connection with spaces of commuting matrices, we need to have linear maps from
a vector space to itself. Define T ∈ A⊗B⊗C to be 1A -generic if there exists α ∈ A∗ with T (α)
invertible. Then T (A∗ )T (α)−1 ⊂ End(B) will be our space of endomorphisms and Strassen’s
Key words and phrases. tensor, commuting matrices, Strassen’s equations, MSC 15.80.
Landsberg partially supported by NSF grant DMS-1006353. Michalek was supported by Iuventus Plus grant
0301/IP3/2015/73 of the Polish Ministry of Science.
1
2
J.M. LANDSBERG AND MATEUSZ MICHALEK
equations for border rank m is that this space is abelian, i.e., in bases we obtain a space of
commuting matrices.
Of particular interest is when an m-dimensional space of commuting matrices, viewed as
a point of the Grassmannian G(m, End(B)), is in the closure of the space of diagonalizable
subspaces (i.e., the maximal tori in gln ), which is denoted Red(m) in [23]. Much of this paper
will utilize the interplay between the tensor and endomorphism perspectives.
One motivation for this paper comes from the study of the complexity of the matrix multipli2
2
2
cation tensor M⟨n⟩ ∈ Cn ⊗Cn ⊗Cn . We initiate a geometric study of the tensors used to prove
upper bounds on the exponent of matrix multiplication, especially the Coppersmith-Winograd
tensor. We identify its special geometric properties and describe other tensors with similar geometric properties in the hope of proving further upper bounds. We plan to discuss other tensors
with these properties and their value (in the sense of [43, 38, 32, 2]) in future work.
Another motivation from computer science is the construction of explicit tensors of high rank
and border rank, see, e.g., [1, 36]. We give several such examples.
Our results include
● Two purely geometric characterizations of the Coppersmith-Winograd tensor (Theorems
7.3 and 7.4).
● Determination of the ranks of numerous tensors of minimal border rank including: all
1A -generic tensors that satisfy Strassen’s equations for m = 4 and m = 5.
● Proof, in §6, when m ≤ 4, of a conjecture of J. Rhodes [3, Conjecture 0] for tensors in
Cm ⊗Cm ⊗Cm that their ranks cannot be twice their border ranks, and counter-examples
for all m > 4 (Proposition 6.5).
● Explicit examples of tensors with rank to border rank ratio greater than two (Proposition
6.10 and Theorem 6.11).
● The flag algebras of [23] are of minimal border rank, §6.2.
● The various necessary conditions for minimal border rank are independent, shown with
explicit examples, §5.
● 1-generic tensors satisfying Strassen’s equations can still be far from minimal border
rank, §5.
● 1-generic tensors satisfying Strassen’s equations must be symmetric, Proposition 5.8.
● New necessary conditions for border rank to be minimal, Theorem 2.4 with an example,
Example 5.6, answering a question of [33].
● A class of tensors for which Strassen’s additivity conjecture holds, Theorem 4.1.
1.1. Background and previous work. The maximum rank of T ∈ Cm ⊗Cm ⊗Cm is not known,
it is easily seen to be at most m2 , and of course is at least the maximum border rank. The
m3 −1
⌉ except when m = 3 when it is five [34, 39]. In computer science,
maximum border rank is ⌈ 3m−2
there is interest in producing explicit tensors of high rank and border rank. The maximal rank
of a known explicit tensor is 3m − log2 (m) − 3 when m is a power of two [1], see Example 3.3
below.
Tensors in A⊗B⊗C are completely understood when all vector spaces have dimension at
most three, see [11]. In particular, for tensors of border rank three, the maximum rank is five.
The case of C2 ⊗Cm ⊗Cm is also completely understood, see, e.g. [27, §10.3]. While tensors of
border rank 4 in C4 ⊗C4 ⊗C4 are essentially understood [18, 19, 4], the ranks of such tensors are
not known. We determine their ranks under our two genericity hypotheses. The difficulty of
understanding border rank four tensors in C4 ⊗C4 ⊗C4 (which was first overcome for this case in
[18]) was non-concision, which we avoid in this paper.
ABELIAN TENSORS
3
Let Red0,SL (m) denote the set of all maximal tori in SL(m), i.e., the set of all (m − 1)dimensional abelian subgroups that are diagonalizable. It can be given a topology (called the
Chabauty topology, see [33]) and its closure RedSL (m) is studied in [33]. (A. Leitner works
over R, but this changes little.) If one considers the corresponding Lie algebras, one obtains a
subvariety of the Grassmannian Redsl (m) = Red(m) ⊂ G(m−1, slm ) that was studied classically,
and is called the variety of reductions in [23]. One can equivalently prove results at the Lie group
or Lie algebra level.
We present the results of [33] (some of which we had found independently) in tensor language
for the benefit of the tensor community.
A tensor T ∈ A⊗B⊗C is A-concise if the map T ∶ A∗ → B⊗C is injective, and it is concise if
it is A, B and C concise. Equivalently, T is A-concise if it does not lie in any A′ ⊗B⊗C with
A′ ⊊ A. Note that if T is A-concise, then R(T ) ≥ a.
Definition 1.1. If b = c = m define T ∈ A⊗B⊗C to be 1A -generic if T (A∗ ) contains an element
of rank m. Define 1B , 1C genericity similarly and say T is 1-generic if it is 1A , 1B and 1C -generic.
Note that if T is 1A -generic, then T is B and C concise and in particular, R(T ) ≥ m.
1.2. Organization. In §2 we describe necessary conditions for 1A -generic tensors to have border
rank m. In addition to Strassen’s equations, there is an End-closed condition, flag genericity
conditions, and infinitesimal flag genericity conditions, the last of which is new. In §3, we
describe the method of [1] for proving lower bounds on the ranks of explicit tensors. This
method has a consequence for the study of Strassen’s additivity conjecture that we describe
in §4. In §5 we study 1A -generic tensors satisfying Strassen’s equations that have border rank
greater than m, giving explicit examples where each of the necessary conditions fail and showing
that such tensors can have very large border rank. Moreover, we show that a 1-generic tensor
satisfying Strassen’s equations is isomorphic to a symmetric tensor. In §6 we study 1A -generic
tensors of minimal border rank, presenting a sufficient condition to have minimal border rank,
classifications when m = 4, 5, computing the ranks as well, and explicit examples of tensors
with large gaps between rank and border rank. We conclude in §7 with a geometric analysis
of tensors that have been useful for proving upper bounds on the complexity of the matrix
multiplication tensor, in particular, giving two geometric characterizations of the CoppersmithWinograd tensor.
1.3. Notation. Let V be a complex vector space, V ∗ = {α ∶ V → C ∣ α is linear} denotes
the dual vector space, V ⊗k denotes the k-th tensor power, S k V denotes the symmetric tensors
in V ⊗k , equivalently, the homogeneous polynomials of degree k on V ∗ , and Λk V denotes the
skew-symmetric tensors in V ⊗k .
Projective space is PV = (V /0)/C∗ . For v ∈ V , [v] ∈ PV denotes the corresponding point in
projective space and for any subset Z ⊂ PV , Ẑ ⊂ V is the corresponding cone in V . For a variety
X ⊂ PV , Xsmooth denotes its smooth points. For x ∈ Xsmooth , T̂x X ⊂ V denotes its affine tangent
space. For a subset Z ⊂ V or Z ⊂ PV , its Zariski closure is denoted Z.
The irreducible polynomial representations of GL(V ) are indexed by partitions π = (p1 , ⋯, pq )
with at most dim V parts. Let `(π) denote the number of parts of π, and let Sπ V denote the
irreducible GL(V )-module corresponding to π. The conjugate partition to π is denoted π ′ .
Since we lack systematic methods to prove bounds on the ranks of tensors, we often rely on
the presentation of a tensor in a given basis to help us. For example, the structure tensor for
4
J.M. LANDSBERG AND MATEUSZ MICHALEK
the group algebra of Zm in the standard basis looks like
x1 ⋯ xm−1 ⎞
⎛ x0
⎜xm−1 x0 x1
⋯ ⎟
⎟ ∣ xj ∈ C}
MC[Zm ] (A∗ ) = {⎜
⎜ ⋮
⎟
⋱
⎜
⎟
⎝ x1
x2 ⋯ x0 ⎠
but after a change of basis (the discrete Fourier transform), it becomes diagonalized so in the
new basis it is transparently of rank and border rank m.
For T ∈ A⊗B⊗C, introduce the notation for T (A∗ ) omitting the xj ∈ C, e.g. for MC[Zm ] (A∗ ),
we write
(1)
x1 ⋯ xm−1 ⎞
⎛ x0
x
x
⋯ ⎟
⎜
0 x1
⎟.
MC[Zm ] (A∗ ) = ⎜ m−1
⎜ ⋮
⎟
⋱
⎝ x1
x2 ⋯
x0 ⎠
For tensors T, T ′ ∈ Cm ⊗Cm ⊗Cm = A⊗B⊗C, we will say T and T ′ are strictly isomorphic
if there exists g ∈ GL(A) × GL(B) × GL(C) such that g(T ) = T ′ , and we will say T, T ′ are
isomorphic if there exists g ∈ GL(A) × GL(B) × GL(C) and σ ∈ S3 such that σ(g(T )) = T ′ .
1.4. Acknowledgments. We thank the Simons Institute for the Theory of Computing, UC
Berkeley, for providing a wonderful environment during the program Algorithms and Complexity
in Algebraic Geometry during which work for this article began. We also thank L. Manivel for
useful discussions and pointing out the reference [33]. Michalek is a member of AGATES group
and a PRIME DAAD fellow.
2. Border rank m equations for tensors in Cm ⊗Cm ⊗Cm
2.1. Strassen’s commutativity equations. Given T ∈ A⊗B⊗C and α ∈ A∗ , consider T (α) ∈
B⊗C = Hom(C ∗ , B). If dim B = dim C = m and T (α) is invertible, for all α′ ∈ A∗ , we may
consider T (α′ )T (α)−1 ∶ B → B.
Strassen’s equations [39] are: for all α, α1 , α2 ∈ A∗ with T (α) invertible,,
rank[T (α1 )T (α)−1 , T (α2 )T (α)−1 ] ≤ 2(R(T ) − m).
In particular, if R(T ) = m, then the space T (A∗ )T (α)−1 ⊂ End(B) is abelian. It is also useful
to use Ottaviani’s formulation of Strassen’s equations [35]: consider the map
TA∧ ∶ B ∗ ⊗A → Λ2 A⊗C
β⊗a ↦ a ∧ T (β).
If dim A = 3, R(T ) ≥ 12 rank(TA∧ ). If one restricts T to a 3-dimensional subspace of A∗ , the same
conclusion holds. In general rank(TA∧ ) ≤ (a − 1)R(T ), because for a rank one tensor a⊗b⊗c,
(a⊗b⊗c)∧A (A⊗B ∗ ) = a ∧ A⊗c, i.e. rank(a⊗b⊗c)∧A = a − 1.
To deal with the case where T (α) is not invertible, recall that a linear map f ∶ B → C ∗ , induces
linear maps f ∧k ∶ Λk B → Λk C ∗ , and that Λm−1 B ≃ B ∗ ⊗Λm B. Thus f ∧(m−1) ∶ Λm−1 B → Λm−1 C ∗
may be identified with (up to a fixed choice of scale) a linear map B ∗ → C, and thus its
transpose may be identified with a linear map C ∗ → B. If f is invertible, this linear map
coincides up to scale with the inverse. In bases it is given by the cofactor matrix of f . So
to obtain polynomials, use (T (α)∧m−1 )T ∶ Λm−1 C → Λm−1 B ∗ in place of T (α)−1 by identifying
Λm−1 C ≃ C ∗ , Λm−1 B ∗ ≃ B, see [27, §3.8.4] for details.
ABELIAN TENSORS
5
As a module, as observed in [29], Strassen’s degree m + 1 A-equations are
(2)
Sm−1,1,1 A∗ ⊗S2,1m−1 B ∗ ⊗S2,1m−1 C ∗ .
2.2. The flag condition. Much of this paper will use the fact that T ∈ A⊗B⊗C may be
recovered up to strict isomorphism from the linear space T (A∗ ) ⊂ B⊗C, and if dim B = dim C
and there exists α ∈ A∗ with T (α) invertible, T may be recovered up to isomorphism from the
space T (A∗ )T (α)−1 ⊂ End(B). In this regard, we recall:
Proposition 2.1. [27, Cor. 2.2] There exist r rank one elements of B⊗C such that T (A∗ ) is
contained in their span if and only if R(T ) ≤ r. Similarly, R(T ) ≤ r if and only if there exists a
curve Et in the Grassmannian G(r, B⊗C), where for t ≠ 0, Et is spanned by r rank one elements
and T (A∗ ) ⊂ E0 (which is defined by the compactness of the Grassmannian).
Corollary 2.2. Let a = m and let T ∈ A⊗B⊗C be A-concise. Then R(T ) = m implies that
T (A∗ ) ∩ Seg(PB × PC) ≠ ∅.
Proof. If Et ∈ G(r, B⊗C) is spanned by rank one elements for all t ≠ 0, then when t = 0, it must
contain at least one rank one element.
More generally, the following result is a slight generalization of [33, Cor. 18], because it does
not assume T is 1A -generic:
Corollary 2.3. Let T ∈ Cm ⊗Cm ⊗Cm = A⊗B⊗C. If R(T ) = m, then there exists a complete
flag A1 ⊂ ⋯Am−1 ⊂ Am = A∗ , with dim Aj = j, such that PT (Aj ) ⊂ σj (Seg(PB × PC)).
Proof. Write T (A∗ ) = limt→0 span{X1 (t), ⋯, Xm (t)} where Xj (t) ∈ B⊗C have rank one. Then
take PAk = P limt→0 span{X1 (t), ⋯, Xk (t)} ⊂ PT (A∗ ). Since P{X1 (t), ⋯, Xk (t)} ⊂ σk (Seg(PB ×
PC)) the same must be true in the limit, and each limit must have the correct dimension because
dim limt→0 span{X1 (t), ⋯, Xm (t)} = m.
There are infinitesimal and scheme-theoretic analogs of Corollary 2.3, but we were unable to
state them in general in a useful manner. Here is a special case that indicates the general case.
For another example, see §7.4. For a variety X ⊂ PV , and a smooth point x ∈ X, T̂x X ⊂ V
denotes its affine tangent space.
Proposition 2.4. Let T ∈ Cm ⊗Cm ⊗Cm = A⊗B⊗C. If R(T ) = m and T (A∗ ) ∩ Seg(PB × PC) =
[X0 ] is a single point, then T (A∗ ) ∩ T̂[X0 ] Seg(PB × PC) must contain a line.
Proof. Say T (A∗ ) were the limit of span{X1 (t), ⋯, Xm (t)} with each Xj (t) of rank one. Then
since PT (A∗ ) ∩ Seg(PB × PC) = [X0 ], we must have each Xj (t) limiting to X0 . But then
limt→0 span{X1 (t), X2 (t)}, which must be two-dimensional, must be contained in T̂[X0 ] Seg(PB ×
PC) and T (A∗ ).
Remark 2.5. Because these conditions deal with intersections, they are difficult to write down as
polynomials. We will use them for tensors with simple expressions where they can be checked.
2.3. Review and clarification of results in [29]. To a 1A -generic tensor T ∈ A⊗B⊗C, fixing
α0 ∈ A∗ as in Definition 1.1, associate a subspace of endomorphisms of B:
Eα0 (T ) ∶= {T (α)T (α0 )−1 ∣ α ∈ A∗ } ⊂ End(B).
Note that T may be recovered up to isomorphism from Eα0 (T ).
Lemma 2.6. Let T ∈ A⊗B⊗C be 1A -generic and assume rank(T (α0 )) = m.
(1) If R(T ) = m then Eα0 (T ) is commutative.
(2) If Eα0 (T ) is commutative then Eα′0 (T ) is commutative for any α0′ ∈ A∗ such that rank(T (α0′ )) =
m.
6
J.M. LANDSBERG AND MATEUSZ MICHALEK
Proof. The first assertion is just a restatement of Strassen’s equations. For the second, say
Eα0 (T ) is commutative, so
(3)
T (α1 )T (α0 )−1 T (α2 ) = T (α2 )T (α0 )−1 T (α1 )
for all α1 , α2 ∈ A∗ . We need to show that
(4)
T (α1 )T (α0′ )−1 T (α2 ) = T (α2 )T (α0′ )−1 T (α1 )
for all α1 , α2 ∈ A∗ . Since Eα0 (T ) is commutative, we have T (α0′ )T (α0 )−1 T (α2 ) = T (α2 )T (α0 )−1 T (α0′ )
and T (α0′ )T (α0 )−1 T (α1 ) = T (α1 )T (α0 )−1 T (α0′ ), i.e., assuming T (α1 ), T (α2 ) are invertible,
T (α0 )−1 = T (α0′ )−1 T (αj )T (α0 )−1 T (α0′ )T (αj )−1 for j = 1, 2.
Substituting the j = 2 case to the left hand side of (3) and the j = 1 case to the right hand side
yields (4). The cases where T (αj ) are not invertible follow by taking limits, as the α with T (α)
invertible form a Zariski open subset of T (A∗ ).
If U ⊂ B ∗ ⊗B is commutative, then we may consider it as an abelian Lie-subalgebra of gl(B).
Define
AbelA ∶ = P{T ∈ A⊗B⊗C ∣ T is A − concise,
∃α0 ∈ A∗ with rank(T (α0 )) = m, and Eα0 (T ) ⊂ gl(B) is an abelian Lie algebra}
= P{T ∈ A⊗B⊗C ∣ T is A − concise, 1A − generic and
∀α ∈ A∗ with rank(T (α)) = m, Eα (T ) ⊂ gl(B) is an abelian Lie algebra}
Definition 2.7. We say T ∈ A⊗B⊗C is an A-abelian tensor if T ∈ AbelA .
AbelA is a Zariski closed subset of the set of concise 1A -generic tensors, namely the zero set
of Strassen’s equations. Its closure in A⊗B⊗C is a component of the zero set of Strassen’s
equations.
Define
0
∶ = P{T ∈ A⊗B⊗C ∣ T is A − concise,
DiagA
∃α0 ∈ A∗ with rank(T (α)) = m, and Eα0 (T ) ⊂ gl(B) is diagonalizable},
= P{T ∈ A⊗B⊗C ∣ T is A − concise, 1A − generic and
∀α ∈ A∗ with rank(T (α)) = m, Eα (T ) ⊂ gl(B) is diagonalizable}.
g
0
Let DiagA be the Zariski closure of DiagA
and let DiagA
be the intersection of DiagA with
the set of concise 1A -generic tensors.
Proposition 2.8. [29] Let A, B, C = Cm , let T ∈ A⊗B⊗C be concise, 1A -generic and satisfy the
A-Strassen equations, i.e., T ∈ AbelA . Then the following are equivalent:
(1) R(T ) = m,
g
(2) T ∈ DiagA
.
Moreover, an abelian m-dimensional subspace of gl(B) is in the closure of diagonalizable subspaces if and only if it arises as Eα (T ) for some concise, border rank m tensor T ∈ A⊗B⊗C.
Proof. Since the proof in the literature is not explicit and we use it frequently, we show that if
T ∈ A⊗B⊗C is concise, then Eα (T ) belongs to the limit of diagonalizable subalgebras. We know
there exists a sequence of m-tuples of rank one tensors (Tit )m
i=1 such that in the Grassmannian
lim span{Tit (A∗ )} → T (A∗ ).
t→0
ABELIAN TENSORS
7
In particular, there exist Xt ∈ span{Tit (A∗ )}, such that Xt → T (α) and we may assume that Xt
are invertible. Then, span{Tit (A∗ )}Xt−1 is a sequence of diagonalizable algebras converging to
Eα (T ).
2.4. The End-closed condition. Define
End −AbelA ∶ = P{T ∈ AbelA ∣ ∃α ∈ A∗ with rank(T (α)) = m, and Eα (T ) is closed under composition}
= P{T ∈ AbelA ∣ ∀α ∈ A∗ with rank(T (α)) = m, Eα (T ) is closed under composition}
If T ∈ End −AbelA , we will say T is End-closed.
Remark 2.9. There was ambiguity in the definition of Comma,b in [29] that is clarified by the
above notions which replace it.
The following Proposition essentially dates back to Gerstenhaber [20]. It is utilized in [33, §5]
to obtain explicit abelian subspaces that are not in Red(m), see §5.1.
Proposition 2.10. If T ∈ Cm ⊗Cm ⊗Cm is 1A -generic, R(T ) = m and rank(T (α0 )) = m, then
Eα0 (T ) is closed under composition.
Proof. Each diagonalizable Lie algebra is a subalgebra of the algebra of matrices. The property
remains true in the closure.
Corollary 2.11. Let T ∈ Cm ⊗Cm ⊗Cm = A⊗B⊗C be of border rank m and assume rank(T (α0 )) =
m. Then Eα0 (T ) = Eα′0 (T ) for any α0′ ∈ A∗ such that rank(T (α0′ )) = m.
Proof. First, the inverse of each invertible element of Eα0 (T ) also belongs to Eα0 (T ) because
this is true for diagonalizable algebras and if sn → s, with s invertible (so we may take each sn
−1
invertible), then s−1
n →s .
The endomorphism T (α0′ )T (α0 )−1 is an invertible element of Eα0 (T ), as it is a composition
of two isomorphisms. Hence, T (α0 )T (α0′ )−1 ∈ Eα0 (T ). By Proposition §2.10 we have:
Eα0 (T ) = T (A∗ )T (α0 )−1 = T (A∗ )T (α0 )−1 T (α0 )T (α0′ )−1 = Eα′0 (T ).
Note the inclusions
g
DiagA
⊆ End −AbelA ⊆ AbelA .
g
These spaces all coincide when m ≤ 4, DiagA
= End −AbelA for m = 5, End −AbelA ⊊ AbelA when
m ≥ 5 (§5.1), and are all different when m ≥ 7 (§5.2).
In addition to providing equations beyond Strassen’s for border rank m, Proposition 2.8 can
be used to derive information on diagonalizable algebras, see, e.g., §6.2.
Proposition 2.12. The subvariety End −AbelA in the set of 1A -generic tensors has equations
that as a GL(A) × GL(B) × GL(C)-module include
Sm,3,1m−2 A∗ ⊗(
⊕
Sπ+(1m ) B ∗ ⊗Sπ′ +(1m ) C ∗ ).
∣π∣=m+1
p1 ,`(π)≤m
These equations are of degree 2m + 1.
Proof. For α1 , ⋯, αm a basis of A∗ , if T ∈ End −AbelA , then for all α, α′ ∈ A∗ ,
T (α)(T (α1 )∧m−1 )T T (α′ ) ⊂ span{T (α1 ), ⋯, T (αm )}.
In other words, the following vector in Λm+1 (B⊗C) must be zero:
T (α)(T (α1 )∧m−1 )T T (α′ ) ∧ T (α1 ) ∧ ⋯ ∧ T (αm ).
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J.M. LANDSBERG AND MATEUSZ MICHALEK
The entries of this vector are polynomials of degree 2m + 1 in the coefficients of T , as the entries
of (T (α1 )∧m−1 )T are of degree m − 1 in the coefficients of T and all the other matrices have
entries that are linear in the coefficients of T . Among the quantities that must be zero are
the coefficients of b1 ⊗c1 ∧ ⋯ ∧ b1 ⊗cm ∧ b2 ⊗c1 , and more generally the coefficients of b1 ⊗c1 ∧ ⋯ ∧
b1 ⊗cqp1 ∧ ⋯ ∧ bp1 ⊗c1 ∧ ⋯ ∧ bp1 ⊗cqp1 where π = (p1 , ⋯, pq1 ) is a partition of m + 1 with first part at
most m, q1 ≤ m, and π ′ = (q1 , ⋯, qp1 ). Now take α, α′ = α2 , the corresponding coefficients have
the stated weight and all are highest weight vectors.
3. The Alexeev-Forbes-Tsimerman method for bounding tensor rank
Because the set of tensors of rank at most r is not closed, there are few techniques for proving
lower bounds on rank that are not just lower bounds for border rank. What follows is the only
general technique we are aware of. (However for very special tensors like matrix multiplication,
additional methods are available, see [28].) The method below, generally called the substitution
method was found independently by several authors. We follow the novel application of it from
[1]. Fix a basis a1 , ⋯, aa of A. Write T = ∑ai=1 ai ⊗ Mi , where Mi ∈ B ⊗ C.
Proposition 3.1. [1, Appendix B], [6, Chapter 6] Let R(T ) = r and M1 ≠ 0. Then there exist
constants λ2 , . . . , λm , such that the tensor
m
T̃ ∶= ∑ aj ⊗ (Mj − λj M1 ) ∈ a1 ⊥ ⊗B⊗C,
j=2
has rank at most r−1. Moreover, if rank(M1 ) = 1 then for any choices of λj we have R(T̃ ) ≥ r−1.
The statement of Proposition 3.1 is slightly different from the original statement in [1], so we
give a modified proof:
Proof. By Proposition 2.1 there exist rank one homomorphisms X1 , . . . , Xr and scalars dij such
that:
r
Mj = ∑ dij Xi .
i=1
Since M1 ≠ 0 we may assume d11 ≠ 0 and define λj =
d1j
.
d11
Then the subspace T̃ (a1 ⊥ ) is spanned
by X2 , . . . , Xr so Proposition 2.1 implies R(T̃ ) ≤ r − 1. The last assertion holds because if
rank(M1 ) = 1 then we may assume X1 = M1 .
Proposition 3.1 is usually implemented by consecutively applying the following steps, which
we will refer to as the substitution method:
(1) Distinguish A, take a basis {aj } of it and take bases {βi }, {γj } of B ∗ , C ∗ and represent
T as a matrix M with entries that are linear combinations of the basis vectors ai :
Mi,j = T (βi ⊗ γj ).
(2) Choose a subset of b′ columns of M and c′ rows of M .
(3) Inductively, for elements of the chosen columns (resp. rows) remove the u-th column
(resp. row) and add to all other columns (resp. rows) the u-th column (resp. row) times
an arbitrary coefficient λ, regarding the aj as formal variables. This step is just to ensure
that each time only nonzero columns or rows are removed.
(4) Set all aj that appeared in any of the selected rows or columns to zero, obtaining a
matrix M ′ . Notice, that M ′ does not depend on the choice of λ.
(5) The rank of T is at least b′ plus c′ plus the rank of the tensor corresponding to M ′ .
The above steps can be iterated, interchanging the roles of A, B and C.
ABELIAN TENSORS
9
Example 3.2. [1] Let
⎛x1
⎞
x1
⎜
⎟
⎜
⎟
⎜
⎟
x
1
⎜
⎟
⎜
⎟
x
1
∗
⎜
⎟.
T (A ) = ⎜
⎟
x1
⎜x2
⎟
⎜
⎟
x2
x1
⎜
⎟
⎜
⎟
⎜x3
⎟
x2
x1
⎝x4 x3
x2
x1 ⎠
Then R(T ) ≥ 15. Indeed, in the first iteration of the method presented above, choose the first
four rows and last four columns. One obtains a 4 × 4 matrix M ′ and the associated tensor T ′ ,
so R(T ) ≥ 8 + R(T ′ ). Iterating the method two times yields R(T ) ≥ 8 + 4 + 2 + 1 = 15.
On the other hand R(T ) = 8, e.g., because T (A∗ ), after a choice of α1 , is a specialization of
a space that is abelian and contains a regular nilpotent element (see Corollary 6.2 below).
To see that R(T ) = 15, one can construct an explicit expression or appeal to Proposition 6.3
because T (A∗ ) is a degeneration of the centralizer of a regular nilpotent element.
k
k
This generalizes to T ∈ Ck ⊗C2 ⊗C2 of rank 2 ∗ 2k − 1 and border rank 2k .
Example 3.3. [1] Let T = a1 ⊗(b1 ⊗c1 +⋯+b8 ⊗c8 )+a2 ⊗(b1 ⊗c5 +b2 ⊗c6 +b3 ⊗c7 +b4 ⊗c8 )+a3 ⊗(b1 ⊗c7 +
b2 ⊗c8 ) + a4 ⊗b1 ⊗c8 + a5 ⊗b8 ⊗c1 + a6 ⊗b8 ⊗c2 + a7 ⊗b8 ⊗c3 + a8 ⊗b8 ⊗c4 , so
⎛x1
x1
⎜
⎜
⎜
x1
⎜
⎜
x1
T (A∗ ) = ⎜
⎜x2
x1
⎜
⎜
x2
x1
⎜
⎜
⎜x3
x2
x1
⎝x4 x3
x2
x5 ⎞
x6 ⎟
⎟
x7 ⎟
⎟
x8 ⎟
⎟.
⎟
⎟
⎟
⎟
⎟
⎟
x1 ⎠
Then R(T ) ≥ 18. Here we start by distinguishing the spaces A and B, contracting a5 , a6 , a7 , a8 .
We obtain a tensor T̃ represented by the matrix
⎛x1
⎞
x1
⎜
⎟
⎜
⎟
⎜
⎟
x
1
⎜
⎟
⎜
⎟
x
1
⎜
⎟,
⎜x2
⎟
x1
⎜
⎟
⎜
⎟
x2
x1
⎜
⎟
⎜
⎟
⎜x3
x2
x1 ⎟
⎝x4 x3
⎠
x2
and R(T ) ≥ 4 + R(T̃ ). The substitution method then gives R(T̃ ) ≥ 14. This generalizes to
k
k
k
T ∈ C2 ⊗C2 ⊗C2 of rank 3 ∗ 2k − k − 3. In fact, R(T ) = 18; it is enough to consider 17 matrices
with just one nonzero entry corresponding to all nonzero entries of T (A∗ ), apart from the top
left and bottom right corner and 1 matrix with 1 at each corner and all other entries equal to 0.
For tensors in Cm ⊗Cm ⊗Cm , the limit of the method would be to prove a tensor has rank at
least 3(m−1), and this can be achieved only by exchanging the roles of A, B, C in the application
successively.
10
J.M. LANDSBERG AND MATEUSZ MICHALEK
4. A Remark on Strassen’s additivity conjecture
Strassen’s additivity conjecture [41] states that the rank of the sum of two tensors in disjoint
spaces equals the sum of the ranks. While this conjecture has been studied from several different
perspectives, e.g. [16, 25, 8, 13, 9], very little is known about it, and experts are divided as to
whether it should be true or false.
In many cases of low rank the substitution method provides the correct rank. In light of this,
the following theorem indicates why providing a counter-example to Strassen’s conjecture may
be difficult.
Theorem 4.1. Let T1 ∈ A1 ⊗B1 ⊗C1 and T2 ∈ A2 ⊗B2 ⊗C2 be such that that R(T1 ) can be
determined by the substitution method. Then Strassen’s additivity conjecture holds for T1 ⊕ T2 ,
i.e, R(T1 ⊕ T2 ) = R(T1 ) + R(T2 ).
Proof. With each application of the substitution method, T1 is modified to a tensor of lower
rank living in a smaller space and T2 is unchanged. After all applications, T1 has been modified
to zero and T2 is still unchanged.
The rank of any tensor in C2 ⊗B⊗C can be computed using the substitution method as follows:
by dimension count, we can always find either β ∈ B ∗ or γ ∈ C ∗ , such that T (β) or T (γ) is a
rank one matrix. In particular, Theorem 4.1 provides an easy proof of Strassen’s additivity
conjecture if the dimension of any of A1 , B1 or C1 equals 2. This was first shown in [25] by other
methods.
5. Abelian tensors in Cm ⊗Cm ⊗Cm with border rank greater than m
5.1. End −AbelA ⊊ AlgA for m ≥ 5. The lower bounds for border rank in the following two
propositions appeared in [33, Def. 16] in the language of groups. It answers [24, Question A]
and provides an example asked for in [24, Remark after Question B] and a concise 1-generic
tensor T ∈ C5 ⊗C5 ⊗C5 with R(T ) = 6 satisfying Strassen equations.
Proposition 5.1. Let TLeit,5 = a1 ⊗(b1 ⊗c1 + b2 ⊗c2 + b3 ⊗c3 + b4 ⊗c4 + b5 ⊗c5 ) + a2 ⊗(b1 ⊗c3 + b3 ⊗c5 ) +
a3 ⊗b1 ⊗c4 + a4 ⊗b2 ⊗c4 + a5 ⊗b2 ⊗c5 , which gives rise to the linear space
⎛x1
⎞
x1
⎜
⎟
⎜
⎟
x1
⎟.
TLeit,5 (A∗ ) = ⎜x2
⎜
⎟
⎜x3 x4
⎟
x1
⎝
x5 x2
x1 ⎠
Then Eα1 (TLeit,5 ) is an abelian Lie algebra, but not End-closed. I.e., TLeit,5 ∈ AbelA but TLeit,5 ∈/
End −AbelA . In particular, TLeit,5 ∈/ DiagA so R(TLeit,5 ) > 5. In fact, R(TLeit,5 ) = 6 and
R(TLeit,5 ) = 9.
Proof. The first statements are verifiable by inspection. The fact that the border rank of the
tensor is at least 6 follows from Theorem 2.8. The fact that border rank equals 6 follows by
considering rank one matrices:
⎛
⎜
⎜
X1 = ⎜
⎜
⎜1 1
⎝1 1
⎞
⎛
⎟
⎜
⎟
⎜
⎟ , X2 = ⎜ ⎟
⎜
⎟
⎜
⎠
⎝1
2
2
⎞
⎛
⎟
⎜
⎟
⎜
⎟ , X3 = ⎜
⎟
⎜
⎟
⎜
⎠
⎝1
4 ⎞
⎟
⎟
⎟,
⎟
⎟
2⎠
ABELIAN TENSORS
⎛
⎜
⎜
X4 = ⎜ 2
⎜
⎜ 1 ⎝
⎞
⎛
⎟
⎜
⎟
⎜
⎟ , X5 = ⎜
⎟
⎜
⎟
⎜1
⎠
⎝
− 2
11
⎞
⎛ 2
⎟
⎜ ⎟
⎜
⎟ , X6 ⎜ −
⎟
⎜
⎟
⎜
⎠
⎝ 1
⎞
⎟
⎟
⎟.
⎟
⎟
⎠
Then
1
+ X2 + X3 + X4 + X5 + X6 ) + a2 ⊗(X2 − X3 ) + a3 ⊗X5 + a4 ⊗X4 + a5 ⊗X6 ,
which is a sum of six rank one tensors. In terms of tensor products,
1
TLeit,5 = lim 2 [ − a1 ⊗(b1 + b2 )⊗(c4 + c5 ) + (a1 + a2 )⊗(b1 + b3 )⊗(c5 + c3 )
→0 + (a1 − a2 )⊗(b1 + 2 b5 )⊗(c5 + 2 c1 ) + (a1 + 2 a4 )⊗(b2 + b3 )⊗(c4 + c3 )
1
a1 ⊗(−X1
→0 2
TLeit,5 = lim
+ (a1 + 2 a3 )⊗(b1 − b3 + 2 b4 )⊗c4 + (a1 + 2 a5 )⊗b2 ⊗(c5 − c3 + 2 c2 )].
Note that the limit base points are a1 ⊗(b1 +b2 )⊗(c4 +c5 ), a1 ⊗b1 ⊗c5 , a1 ⊗b2 ⊗c4 , a1 ⊗b1 ⊗c4 , a1 ⊗b2 ⊗c5
which are five points on a Seg(P0 × P1 × P1 ) ⊂ P3 .
The substitution method shows that R(TLeit,5 ) ≥ 9. To prove equality, consider the 9 rank 1
matrices:
(1) 3 matrices with just one nonzero entry corresponding to x3 , x4 , x5 ,
(2) The six matrices
⎛1
⎜
⎜
⎜−1
⎜
⎜
⎝−1
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟,⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
−1
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟,⎜
⎟ ⎜
⎟ ⎜
1⎠ ⎝
⎞ ⎛
⎟ ⎜ 1
⎟ ⎜
⎟,⎜
⎟ ⎜
1 ⎟ ⎜
⎠ ⎝
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟ , ⎜1
⎟ ⎜
⎟ ⎜
⎠ ⎝1
1
1
T (A∗ ) is contained in the span of these matrices.
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟,⎜ 1
⎟ ⎜
⎟ ⎜
⎠ ⎝−1
−1
1
⎞
⎟
⎟
⎟.
⎟
⎟
⎠
Note that TLeit,5 is neither 1B nor 1C -generic. The example easily generalizes to higher m,
e.g. for m = 7 we could take:
⎛x1
⎞
x1
⎜
⎟
⎜
⎟
⎜
⎟
x
1
⎜
⎟
∗
⎜
⎟.
x1
TLeit,7 (A ) = ⎜x2
⎟
⎜x
⎟
x1
⎜ 3
⎟
⎜
⎟
⎜x4 x7
⎟
x1
⎝
x5 x6 x2
x1 ⎠
Proposition 5.2. The following tensor:
TLeit,6 = a1 ⊗(b1 ⊗c1 +⋯+b6 ⊗c6 )+a2 ⊗(b1 ⊗c2 +b2 ⊗c3 )+a3 ⊗b1 ⊗c5 +a4 ⊗b1 ⊗c6 +a5 ⊗b4 ⊗c5 +a6 ⊗b4 ⊗c6 ,
which gives rise to the abelian subspace:
⎛x1
⎞
⎜x2 x1
⎟
⎜
⎟
x2 x1
⎜
⎟
∗
⎟,
TLeit,6 (A ) = ⎜
⎜
⎟
x
1
⎜
⎟
⎜x3
⎟
x5 x1
⎝x
x6
x1 ⎠
4
12
J.M. LANDSBERG AND MATEUSZ MICHALEK
is not End-closed and satisfies R(TLeit,6 ) = 7 and R(TLeit,6 ) = 11.
Proof. The border rank is at least 7 as TLeit,6 (A∗ ) is not End-closed. The rank is at least 11 by
the substitution method.
To prove that rank is indeed 11 we notice that the 3 × 3 upper square of TLeit,6 (A∗ )
⎞
⎛x1
⎟
⎜x2 x1
⎝
x2 x1 ⎠
represents a tensor of rank at most 4 by considering:
⎛
⎜
⎝
1⎞ ⎛
⎞ ⎛
⎞ ⎛1
⎟ , ⎜1 1 ⎟ , ⎜ 1 −1 ⎟ , ⎜
⎠ ⎝1 1 ⎠ ⎝−1 1 ⎠ ⎝−1
−1⎞
⎟.
1⎠
Apart from this square there are 7 nonzero entries, so the rank is at most 7 + 4 = 11.
To compute the border rank notice that after removing the second row and column we obtain
a tensor of border rank 5 by Proposition 6.4 below. On the other hand the entries in the second
column and row clearly form a border rank 2 tensor. In other words, the tensor corresponding
to
⎛x1
⎞
⎜x2 x7
⎟
⎜
⎟
x2 x1
⎜
⎟
⎜
⎟.
⎜
⎟
x
1
⎜
⎟
⎜x3
⎟
x5 x1
⎝x
x6
x1 ⎠
4
has border rank 7 and TLeit,6 is a specialization if it.
g
5.2. DiagA
⊊ End −AbelA for m ≥ 7.
Proposition 5.3. The tensor corresponding to
⎛x1
x1
⎜
⎜
⎜
⎜
Tend,7 (A∗ ) = ⎜
⎜
⎜
x2 + x7
⎜
⎜
⎜x2
x3
⎝x4
x5
x1
x1
x3 x4 x1
x5 x6
x1
x6 x7
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
x1 ⎠
is End-closed, but has border rank at least 8.
Proof. The fact that it is End-closed follows by inspection. The tensor has border rank at least
8 by Corollary 2.2 as Tend,7 (A∗ ) does not intersect the Segre. Indeed, if it intersected Segre we
would have x1 = x4 = 0, and (x2 + x7 )x2 = 0. If x2 + x7 ≠ 0, then x2 = 0 and x27 = (x2 + x7 )x7 = 0,
which gives a contradiction. If x2 = 0 analogously we obtain x7 = 0 and x3 = x5 = x6 = 0.
The following proposition yields families of End-closed tensors of large border rank for large
m and border rank greater than m as soon as m ≥ 8:
Proposition 5.4. Let T ∈ A⊗B⊗C = Cm ⊗Cm ⊗Cm be such that T = a1 ⊗(b1 ⊗c1 + ⋯bm ⊗cm ) + T ′
where T ′ ∈ A′ ⊗B ′ ⊗C ′ ∶= span{a2 , ⋯, am }⊗span{b1 , ⋯, b⌊ m ⌋ }⊗span{c⌈ m ⌉ , ⋯, cm }. Then R(T ) =
2
2
R(T ′ ) + m.
ABELIAN TENSORS
13
Proof. We have
⎛x1
⋱
⎜
⎜
x1
⎜
T (A∗ ) ⊂ ⎜
⎜ ∗ ⋯ ∗ x1
⎜
⎜⋮
⋮
⋮
⋱
⎝∗ ⋯ ∗
x1
⎞
⎟
⎟
⎟
⎟.
⎟
⎟
⎟
⎠
The lower bound on rank is obtained by substitution method and the upper bound is trivial. 2
Corollary 5.5. A general element of End −AbelA has border rank at least m8 , which is approximately 38 -of the maximal border rank of a general tensor in Cm ⊗Cm ⊗Cm .
Proof. R(T ) ≥ R(T ′ ) and if T ′ ∈ Cm−1 ⊗C⌊ 2 ⌋ ⊗C⌈ 2 ⌉ is general, it will have border greater than
m2
8 .
m
m
5.3. A generic A-abelian, End-closed tensor satisfying Corollary 2.2 has high border
rank.
Proposition 5.6. Let m−1 = k+` with 3(k+`) < k`+5. Then an abelian tensor T ∈ Cm ⊗Cm ⊗Cm
of the form T = M⟨1⟩ ⊕ T ′ with M⟨1⟩ ∈ C1 ⊕ C1 ⊕ C1 and T ′ ∈ Cm−1 ⊗Cm−1 ⊗Cm−1 = A′ ⊗B ′ ⊗C ′
of the form a1 ⊗(b1 ⊗c1 + ⋯ + bm−1 ⊗cm−1 ) + T ′′ with T ′′ ∈ Cm−2 ⊗Ck ⊗C` general, where Ck =
span{b1 , ⋯, bk } and C` = span{ck + 1, ⋯, cm−1 }, has R(T ) > m + 1 despite being 1A -generic,
abelian, End-closed, and such that PT (A∗ ) ∩ Seg(PB × PC) ≠ ∅.
Proof. For general T ′′ the space T ′′ ((Cm−2 )∗ ) ∈ Ck ⊗C` will not intersect σ2 (Seg(Pk−1 × P`−1 )) if
dim PT ′′ (A∗ )+dim σ2 (Seg(Pk−1 ×P`−1 )) < dim P(Ck ⊗C` ), i.e., if [2(k+`−2)−1]+[k+`−1] < k`−1,
i.e., if 3(k + `) < k` + 5.
5.4. A tensor in End −Abel8 of border rank > 8 via Corollary 2.2. Consider (from [33,
Prop. 19])
TLeit,8 =a1 ⊗(b1 ⊗c1 + ⋯ + b8 ⊗c8 ) + a2 ⊗(b4 ⊗c5 + b3 ⊗c6 ) + a3 ⊗(b3 ⊗c5 + b2 ⊗c6 ) + a4 ⊗(b2 ⊗c5 + b1 ⊗c6 )
+ a5 ⊗(b4 ⊗c7 + b3 ⊗c8 ) + a6 ⊗(b3 ⊗c7 + b2 ⊗c8 ) + a7 ⊗(b2 ⊗c7 + b1 ⊗c8 ) + a8 ⊗(b1 ⊗c7 + b4 ⊗c6 )
so
⎛x1
⎜
⎜
⎜
⎜
⎜
TLeit,8 (A∗ ) = ⎜
⎜
⎜
⎜x
⎜ 4
⎜
⎜x8
⎝x7
x1
x1
x4
x3
x7
x6
x3
x2
x6
x5
x1
x2 x1
x8
x1
x5
x1
⎞
⎟
⎟
⎟
⎟
⎟
⎟.
⎟
⎟
⎟
⎟
⎟
⎟
x1 ⎠
Since PT (A∗ ) ∩ Seg(PB × PC) = ∅, R(TLeit,8 ) > 8. In Leitner’s language the bound arises
because the corresponding group does not contain a one parameter subgroup of rank one.
5.5. A-abelian tensors that intersect the Segre and fail to satisfy flag condition. The
following example answers a question about sufficient conditions to be a limit of diagonalizable
groups in [33, p. 10].
14
J.M. LANDSBERG AND MATEUSZ MICHALEK
Example 5.7. Consider Ts9 ∶= M⟨1⟩ ⊕TLeit,8 ∈ C9 ⊗C9 ⊗C9 = A⊗B⊗C. To keep indices consistent
with above, write M⟨1⟩ = a0 ⊗b0 ⊗c0 , so they range from 0 to 8. Note that Ts9 ∩Seg(P8 ×P8 ×P8 ) ≠
∅, and Ts9 is 1A -generic and abelian, however R(Ts9 ) > 9. To see this, note that the flag condition
fails because there is no P1 contained in σ2 (Seg(P8 × P8 )).
5.6. An End-closed tensor satisfying the flag condition but not the infinitesimal flag
condition. Consider
⎛x1
⎞
⎜x0 x1
⎟
⎜
⎟
x1
⎜
⎟
⎜
⎟
⎜
⎟
x
1
⎜
⎟
⎟
x
Tf lagok (A∗ ) ∶= ⎜
1
⎜
⎟
⎜
⎟
x4 x3 x2 x1
⎜
⎟
⎜
⎟
⎜x4
⎟
x3 x2 x8
x1
⎜
⎟
⎜x8
⎟
x7 x6 x5
x1
⎝x7
x6 x5
x1 ⎠
Here PTf lagok (A∗ ) ∩ Seg(P8 × P8 ) = [X0 ], where X0 is
zero elsewhere, and
⎛∗
⎜∗
⎜
T̂[X0 ] Seg(P8 × P8 ) = ⎜∗
⎜
⎜⋮
⎝∗
the matrix with 1 in the (2, 1) entry and
0 ⋯ 0⎞
∗ ⋯ ∗⎟
⎟
0 ⋯ 0⎟
⎟
⎟
⋮
0 ⋯ 0⎠
which only intersects Tf lagok (A∗ ) in [X0 ]. Thus by Proposition 2.4, R(Tf lagok ) > 9.
The the flag condition is satisfied: consider respectively spaces spanned by x0 , x4 , x3 , x2 , x8 , x7 , x6 , x4 .
It straightforward to check that Tf lagok is End-closed.
5.7. 1-generic abelian tensors. In general, the spaces T (A∗ ), T (B ∗ ), T (C ∗ ) can be very different, e.g. if T is 1A -generic and not 1B -generic, or if T is not abelian. The following proposition
is a variant of remarks in [27, §7.7.2] and [18]:
Proposition 5.8. Let T ∈ A⊗B⊗C = Cm ⊗Cm ⊗Cm be 1A and 1B generic and satisfy the AStrassen equations. Then T is isomorphic to a tensor in S 2 Cm ⊗Cm .
In particular:
(1) After making choices of general α ∈ A∗ and β ∈ B ∗ , T (A∗ ) and T (B ∗ ) are GLm isomorphic subspaces of End(Cm ).
(2) If T is 1-generic, then T is isomorphic to a tensor in S 3 Cm .
Proof. Let {ai }, {bj }, {cj } respectively be bases of A, B, C. Write T = ∑ tijk ai ⊗bj ⊗ck . Possibly
after a change of basis we may assume t1jk = δjk and ti1k = δik . Take {αi } the dual basis to {aj }
and identify T (A∗ ) ⊂ End(Cm ) via α1 . Strassen’s A-equations then say
0 = [T (αi1 ), T (αi2 )] = ∑ ti1 jl ti2 lk − ti2 jl ti1 lk ∀i1 , i2 , j, k.
l
Consider when j = 1:
0 = ∑ ti1 1l ti2 lk − ti2 1l ti1 lk = ti2 i1 k − ti1 i2 k ∀i1 , i2 , k,
l
because ti1 1l = δi1 ,l . But this says T ∈ S 2 Cm ⊗Cm .
ABELIAN TENSORS
15
For the last assertion, say LB ∶ B → A is such that IdA ⊗LB ⊗IdC (T ) ∈ S 2 A⊗C and LC ∶ C → A
is such that IdA ⊗IdB ⊗LC ∈ S 2 A⊗B. Then IdA ⊗LB ⊗LC (T ) is in A⊗3 , symmetric in the first
and second factors as well as the first and third. But S3 is generated by two transpositions, so
IdA ⊗LB ⊗LC (T ) ∈ S 3 A.
Thus the A, B, C-Strassen equations, despite being very different modules, when restricted
to 1-generic tensors, all have the same zero sets. Strassen’s equations in the case of partially
symmetric tensors were essentially known to Toeplitz, and in the symmetric case to Aronhold.
Proposition 5.9. There exist 1-generic abelian tensors T ∈ C4k ⊗C4k ⊗C4k that have border
k(k+1)
rank at least 6 . In particular, there exist tensors in C4k ⊗C4k ⊗C4k satisfying the StrassenAronhold equations with R(T ) ≥
k(k+1)
.
6
Proof. We exhibit a family of 4k-dimensional subspaces of S 2 C4k whose associated tensor can
degenerate to any tensor T ′ ∈ S 2 Ck ⊗ Ck−1 . Since the border rank of a general tensor in S 2 Ck ⊗
Ck−1 ⊂ Ck ⊗Ck ⊗Ck−1 must satisfy:
dim σR(T ′ ) (Seg(Pk−1 × Pk−1 × Pk−2 )) > dim P(S 2 Ck ⊗ Ck−1 )) =
(k − 1)k(k + 1)
− 1,
2
and dim σr (Seg(Pk−1 × Pk−1 × Pk−2 )) = r(3k − 4) + r − 1 (by [34]), we obtain:
R(T ) ≥ R(T ′ ) ≥
k(k + 1)
.
6
Represent T as a 4k × 4k symmetric matrix M with entries that are linear functions of 4k
variables. The variables will play four different roles, so we name them accordingly:
(1)
(2)
(3)
(4)
variable z,
variables y1 , . . . , yk−1 ,
variables x1 , . . . , xk ,
variables z1 , . . . , z2k .
They appear in the given order (starting from the top) in the first column of M , ensuring 1B genericity. This also defines the last row of M , ensuring 1C -genericity. The matrix M will be
lower triangular, with the variable z on the diagonal (ensuring 1A -genericity), and z appears
only on the diagonal. The variables z1 , . . . , z2k appear only in the last row and first column as
defined previously.
Write a general tensor T ′ ∈ S 2 Ck ⊗ Ck−1 as T ′ = ∑ alij ei ⊗ ej ⊗ fl , where alij = alji . We use T ′
to define the entries in rows 3k + 1, . . . , 4k − 1 and columns k + 1, . . . , 2k: let ∑ki=1 am
is xi be the
entry in the (3k + m, k + s)-position. As M is symmetric this defines also the entries in rows
2k + 1, . . . , 3k and columns 2, . . . , k.
Apart from entries defined so far, the only remaining nonzero entries belong to a k × k
submatrix of rows from 2k + 1, . . . , 3k and columns k + 1, . . . , 2k: the linear form in the y’s
k−i
∑k−1
i=1 a(k+1−m),s yi is the (2k + m, k + s)-entry.
For example, when k = 3,
16
J.M. LANDSBERG AND MATEUSZ MICHALEK
⎛z
⎞
z
⎜ y1
⎟
⎜y
⎟
z
⎜ 2
⎟
⎜
⎟
⎜x1
⎟
z
⎜
⎟
⎜x2
⎟
z
⎜
⎟
⎜x3
⎟
z
⎜
⎟
∗
⎟.
T (A ) = ⎜
3−l
⎜ z1 ∑i a23i xi ∑i a13i xi ∑l a3−l
⎟
yl ∑l a3−l
yl ∑l a33
yl z
13
23
⎜
⎟
3−l
3−l
⎜ z2 ∑i a22i xi ∑i a12i xi ∑l a3−l
⎟
y
a
y
a
y
z
∑
∑
l 22 l
l 23 l
12 l
⎜
⎟
⎜ z ∑ a2 x ∑ a1 x ∑ a3−l y ∑ a3−l y ∑ a3−l y
⎟
z
⎜ 3
⎟
i 1i i
i 1i i
l 11 l
l 12 l
l 13 l
⎜
⎟
1
1
1
⎜ z4
⎟
a
x
a
x
a
x
z
∑
∑
∑
i
i
i
i
i
i
1i
2i
3i
⎜
⎟
⎜ z5
⎟
z
∑i a21i xi ∑i a22i xi ∑i a23i xi
⎝z
⎠
z
z
z
z
z
x
x
x
y
y
z
6
5
4
3
2
1
3
2
1
2
1
To prove that Strassen’s equations are satisfied, take a matrix M in variables as above and
a matrix M ′ with primed variables, then it is sufficient to show that each entry of the product
matrix M M ′ is symmetric as a bilinear form. This is obvious for:
(1) a product of first 2k rows of M with any column of M ′ ,
(2) a product of rows 2k + 1, . . . , 4k − 1 of M with columns 2, . . . , 4k of M ′ ,
(3) a product of the last row of M with column 1 or columns 2k + 1, . . . , 4k of M ′ .
As all matrices are symmetric it remains to check the assumption for the product of rows
2k + 1, . . . , 4k − 1 of M and the first column of M ′ .
For the row 3k + l we obtain the symmetric linear form ∑j (∑i alji xi )x′j .
k
k−l
k−l
′
′
For the row 2k+n we obtain the symmetric linear form ∑k−1
l=1 (∑i ak+1−n,i xi )yl +∑m=1 (∑l a(k+1−n),m yl )xm .
6. Tensors of border rank m in Cm ⊗Cm ⊗Cm
In this section we present sufficient conditions for tensors to be of minimal border rank and
determine the ranks of several examples of minimal border rank.
6.1. Centralizers of a regular element. Let A = B = C = Cm . An element x ∈ End(B) is
regular if dim C(x) = m, where C(x) ∶= {y ∈ End(B) ∣ [x, y] = 0} is the centralizer of x. We say
x is regular semi-simple if x is diagonalizable with distinct eigenvalues. Note that x is regular
semi-simple if and only if C(x) is diagonalizable.
The following proposition was communicated to us by L. Manivel.
Proposition 6.1. Let U ⊂ End(B) be an abelian subspace of dimension m. If there exists
x ∈ U that is regular, then U lies in the Zariski closure of the diagonalizable m-planes in
G(m, End(B)), i.e., U ∈ Red(m). More generally, if there exist x1 , x2 ∈ U , such that U is their
common centralizer, then U ∈ Red(m).
Proof. Since the Zariski closure of the regular semi-simple elements is all of End(B), for any
x ∈ End(B), there exists a curve xt of regular semi-simple elements with limt→0 xt = x. Consider
the induced curve in the Grassmannian C(xt ) ⊂ G(m, End(B)). Then C0 ∶= limt→0 C(xt ) exists
and is contained in C(x) ⊂ End(B) and since U is abelian, we also have U ⊆ C(x). But if x is
regular, then dim C0 = dim(U ) = m, so limt→0 C(xt ), C0 and U must be equal and thus U is a
limit of diagonalizable subspaces.
ABELIAN TENSORS
17
The proof of the second statement is similar, as a pair of commuting matrices can be approximated by a pair of diagonalizable commuting matrices and diagonalizable commuting matrices
are simultaneously diagonalizable, cf. [23, Proposition 4].
Corollary 6.2. Let T ∈ A⊗Cm ⊗Cm with dim A ≤ m be such that T (A∗ ) contains an element of
rank m and, after using it to embed T (A∗ ) ⊂ glm , it is abelian and contains a regular element.
Then R(T ) = m.
Proposition 6.3. Let T (A∗ /α) ⊂ sl(B) be the centralizer of a regular element of Jordan type
(d1 , ⋯, dq ). Then R(T ) = m and R(T ) = 2(∑qj=1 dj ) − q. In particular, if T (A∗ ) is the centralizer
of a regular nilpotent element, then R(T ) = 2m − 1 and if it is the centralizer of a regular
semi-simple element then R(T ) = m.
Proof. It remains to show the second assertion. The substitution method gives the lower bound
on R(T ) for the regular nilpotent case, and Theorem 4.1 gives the lower bound for the general
case. For the upper bound, it is sufficient to prove the regular nilpotent case. The tensor
MC[Z2m−1 ] (see Equation (1)) specializes to T as follows: consider the space MC[Z2m−1 ] (A∗ ), cut
m
the first ⌊ m
2 ⌋ rows and the last ⌊ 2 ⌋ columns and set all entries appearing above the diagonal in
the remaining matrix to zero. E.g., when m = 2,
⎛x1 x2 x3 ⎞
x x1
x
0
⎜x3 x1 x2 ⎟ → ( 3
)→( 3
).
x
x
x
x
2
3
2
3
⎝x2 x3 x1 ⎠
6.2. The flag algebras of [24]. We answer a question posed in [24, p. 4/p. 5] whether certain
algebras derived from flags belong to Red(m). We start by presenting these algebras. Using
matrix notation, the algebras are given by a partition λ of size ∣λ∣ = m−1, to which we associate a
Young tableau with entries {x2 , ⋯, xm } whose reflection (across a vertical line for the American
presentation and across a diagonal line for the French presentation) we situate in the upper
right hand block of the m × m matrix T (A∗ ) and we fill the diagonal with x1 ’s. For example
λ = (4, 2, 1) gives rise to
(5)
x2 x3 x4
⎛x1
x
x6
⎜
1
⎜
⎜
x1
⎜
⎜
x1
∗
T (A ) = ⎜
⎜
x1
⎜
⎜
x1
⎜
⎜
⎜
x1
⎝
x5 ⎞
x7 ⎟
⎟
x8 ⎟
⎟
⎟
⎟.
⎟
⎟
⎟
⎟
⎟
⎟
x1 ⎠
Proposition 6.4. The abelian Lie algebras associated to flags, defined in [24, p. 4/p. 5] belong
to Red(m), the closure of the diagonalizable algebras.
Proof. By Theorem 2.8 it is enough to prove that the associated tensors are of border rank m.
For the purposes of this proof, it will be convenient to re-order bases such that the x1 ’s occur
18
J.M. LANDSBERG AND MATEUSZ MICHALEK
on the anti-diagonal, and the Young tableau occur with the American presentation:
(6)
x1 ⎞
⎛x2 x3 x4 x5
x
x
x
⎜ 6
⎟
7
1
⎜
⎟
⎜x8
⎟
x
1
⎜
⎟
⎜
⎟
x1
⎟.
T (A∗ ) = ⎜
⎜
⎟
x1
⎜
⎟
⎜
⎟
x1
⎜
⎟
⎜
⎟
⎜
⎟
x1
⎝x1
⎠
Suppose that T (A∗ ) is defined by a partition λ = (k lk , ⋯, 2l2 , 1l1 ) with ∣λ∣ = m − 1. We define
m rank 1 matrices, parametrized by such that T (A∗ ) equals the limit of the span of these
matrices as → 0, considered as a curve in the Grassmannian G(m, gl(B)). The matrices will
belong to five groups.
We label the rows and columns of our matrix by 0, ⋯, m − 1.
1) The first group contains just one matrix with four nonzero entries:
0 ...
⎛ 1
⋮
⎜
⎝m−1 0 . . .
m−1 ⎞
⎟.
0 2m−2 ⎠
0
2) The second group also contains one matrix, with support contained in the `(λ) × k upper
left rectangle. For (i, j) in the rectangle, we fill the entry with i+j and set all other entries to
zero. So the tensor (6) gives
0
1
2
3
⎛ ⎜1 2 3 4
⎜ 2 3 4 5
⎜ ⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎞
⎟
⎟
⎟
⎟
⎟
⎟.
⎟
⎟
⎟
⎟
⎟
⎟
⎠
3) The third group contains m − k − `(λ) matrices. Each matrix corresponds to an entry in
the Young diagram λ that is neither in the zero-th row or column. Notice that the number of
such entries equals the number of anti-diagonal entries of the matrix that are not in the first
k − 1 rows or first (∑ki=1 li ) − 1 columns. Fix a bijection between them. To each such entry of the
Young diagram we associate a rank one matrix with only four nonzero entries. Suppose that
the entry of the Young diagram is the (i0 , j0 ) entry of the matrix and the corresponding entry
of the anti-diagonal is (i1 , j1 = m − 1 − i1 ). The entries are
⎧
i0 +j0
⎪
⎪
⎪
⎪
⎪
m−1
⎪
⎪
⎪ i1 +j0
i
aj = ⎨
⎪
⎪
⎪
i0 +j1
⎪
⎪
⎪
⎪
⎪
⎩ 0
(i, j) = (i0 , j0 )⎫
⎪
⎪
⎪
⎪
(i, j) = (i1 , j1 )⎪
⎪
⎪
⎪
(i, j) = (i1 , j0 )⎬ .
⎪
⎪
(i, j) = (i0 , j1 )⎪
⎪
⎪
⎪
⎪
otherwise ⎪
⎭
ABELIAN TENSORS
19
The tensor (6) has
⎛
⎜ 2
⎜
⎜
⎜ 4
⎜ ⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎞
⎟
⎟
⎟
⎟
⎟
⎟.
⎟
⎟
⎟
⎟
⎟
⎟
⎠
5
7
4) The fourth group contains k − 1 matrices. These correspond to the entries in the 0-th row
of λ but not in the 0-th column. The matrix corresponding to the entry in the i-th column is
defined by
⎫
⎧
i
j=0
⎪
⎪
⎪
⎪
⎪
⎪
i+j
⎪
⎪
⎪
⎪
j
≤
k
−
1
and
(i,
j)
∈
/
Young
diagram
ofλ
⎪
⎪
⎪
⎪
⎪
⎪ i+j
m−1−j
i
i
⎬
aj = ⎨−
aj
has been associated to aj
⎪
⎪
⎪
⎪
m−1
⎪
⎪
j
=
m
−
1
−
i
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
0
otherwise
⎭
⎩
The tensor (6) has
1
⎛ ⎜
⎜
⎜ 3
⎜
⎜ −4
⎜
⎜
⎜
⎜
⎜
⎜
⎜ 7
⎝
⎞
⎟
⎟
⎟
⎟
⎟
⎟,
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
2
3
4
⎞
⎟
⎟
⎟
⎟
⎟
⎟,
⎟
⎟
⎟
⎟
⎟
⎟
⎠
7
3
4
5
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎞
⎟
⎟
⎟
⎟
⎟
⎟.
⎟
⎟
⎟
⎟
⎟
⎟
⎠
7
5) The fifth group, consisting of `(λ) − 1 matrices, is analogous to the fourth with entries
corresponding to rows instead of columns and all entries, apart from the first row, in the `(λ) × k
upper left rectangle equal to zero.
The tensor (6) has
⎛
⎜1
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
−5
⎞
7 ⎟
⎟
⎟
⎟
⎟
⎟,
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎛
⎜
⎜ 2
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
7
⎞
⎟
⎟
⎟
⎟
⎟
⎟.
⎟
⎟
⎟
⎟
⎟
⎟
⎠
Except for the second group, each matrix in each group has a distinguished element in the Young
diagram which, after normalization, is the limit as → 0. Moreover, summing all matrices from
groups 1, 3, 4, 5 and subtracting the matrix from group 2, the limit as → 0 is the identity
matrix.
6.3. Case m = 4. Fix an A-concise, 1A -generic border rank 4 tensor T ∈ A ⊗ B ⊗ C, where
A, B, C ≃ C4 . The contraction T (A) is a 4-dimensional subspace of matrices and we may
assume that it contains the identity. By [24, Prop. 18] we may assume that the space T (A) is
20
J.M. LANDSBERG AND MATEUSZ MICHALEK
one of the 14 types of [24, §3.1], corresponding to orbits of P GL4 . We consider tensors up to
isomorphism.
● One-regular algebras - centralizers of regular elements. There are 5 types, their ranks
are provided by Proposition 6.3. These are O12 , O11 , O10′ , O10′′ , O9 in [24, §3.1].
● Containing a (3, 1) Jordan type non-regular element. There are two types, giving rise to
isomorphic rank 6 tensors. Set theoretically, the intersection with the Segre variety is a
line and a point.
TO8′′ ,O8′
⎛c
⎜0
=⎜
⎜0
⎝0
0
c
0
0
a
b
c
0
0⎞
0⎟
⎟
0⎟
d⎠
● Containing a nilpotent element with Jordan block size 3. There are three types, all
of rank 7, the first one representing a class to which the Coppersmith-Winograd tensor T̃2,CW belongs, see §7.2. In the second case the intersection with the Segre settheoretically is a line.
TO8 = T̃2,CW
⎛c
⎜0
=⎜
⎜0
⎝0
b
c
0
0
a
b
c
d
d⎞
⎛c
0⎟
⎜0
⎟ , T ′′ ′ = ⎜
0⎟ O7 ,O7 ⎜0
⎝0
c⎠
b
c
0
0
a
b
c
d
0⎞
0⎟
⎟,
0⎟
c⎠
● Four types, all of rank 7, giving rise to three different types of tensors. Set-theoretically
the intersection with the Segre is, in the first case two lines intersecting in a point, in
the second case a smooth quadric, in the third case a plane.
TO6
⎛c
⎜0
=⎜
⎜0
⎝0
0
c
0
0
a
0
c
0
b⎞
⎛c
d⎟
⎜0
⎟, T = ⎜
0⎟ O7 ⎜0
⎝0
c⎠
0
c
0
0
a
b
c
0
d⎞
⎛c
a⎟
⎜0
⎟ , T ′ ′′ = ⎜
0⎟ O3 ,O3 ⎜0
⎝0
c⎠
a
c
0
0
b
0
c
0
d⎞
0⎟
⎟.
0⎟
c⎠
6.4. Case m = 5. As remarked in [23], it is sufficient to consider nilpotent subspaces as others
are built out of them, so we restrict our attention to them. Up to transpositions the following
are the only maximal, nilpotent, End-abelian 5-dimensional subalgebras of the algebra of 5 × 5
matrices. We prove that each of them is in Red(5). Notation is such that TNi,j corresponds to
the nilpotent algebras Ni , Nj of [42], and we slightly abuse notation, identifying the tensor with
ABELIAN TENSORS
21
its corresponding linear space.
TN1,4
⎛a
⎜b
⎜
= ⎜c
⎜
⎜d
⎝e
TN10,12
TN15
0
a
0
0
0
⎛a
⎜b
⎜
= ⎜c
⎜
⎜d
⎝e
⎛a
⎜b
⎜
= ⎜c
⎜
⎜d
⎝e
0
0
a
0
0
0
a
b
0
0
0
a
b
c
0
0
0
a
0
0
0
0
a
b
0
0
0
a
0
0
0
e
d
a
0
0⎞
⎛a 0
0⎟
⎜b a
⎜
⎟
e ⎟ , TN7,9 = ⎜ c b
⎜
⎟
⎜0 0
0⎟
⎝0 0
a⎠
0⎞
⎛a 0
0⎟
⎜b a
⎟
⎜
0⎟ , TN11,13 = ⎜ c b
⎟
⎜
⎜d 0
0⎟
⎠
⎝e 0
a
0
0
a
0
0
0
0
d
a
0
0⎞
⎛a 0
0⎟
⎜b a
⎜
⎟
0⎟ , TN6,8 = ⎜ c b
⎜
⎟
⎜0 0
0⎟
⎝e 0
a⎠
0
0
0
a
0
0
0
0
a
0
0
0
0
a
0
0⎞
⎛a 0
0⎟
⎜b a
⎟
⎜
0⎟ , TN16 = ⎜ c b
⎟
⎜
⎜d c
0⎟
⎠
⎝e 0
a
0
0
a
b
0
0
0
0
a
0
0
0
a
0
0
0
d
e
a
b
0⎞
0⎟
⎟
d⎟ ,
⎟
0⎟
a⎠
0⎞
⎛a 0
0⎟
⎜b a
⎟
⎜
0⎟ , TN14 = T̃3,CW = ⎜ c b
⎟
⎜
⎜d 0
0⎟
⎠
⎝e 0
a
0⎞
⎛a 0
0⎟
⎜b a
⎟
⎜
0⎟ , TN17 = ⎜ c b
⎟
⎜
⎜d c
e⎟
⎠
⎝0 0
a
0
0
a
b
0
0
0
0
a
0
0
0
a
0
0
0
0
d
a
0
0⎞
0⎟
⎟
e⎟ ,
⎟
0⎟
a⎠
0⎞
0⎟
⎟
0⎟ .
⎟
e⎟
a⎠
TN1,4 is obviously of border rank five. For TN6,8 ,TN15 ,TN16 , and TN17 (resp. TN7,9 ) apply Proposition 6.1 to a pair of matrices represented by b and e (resp. b,d). TN10,12 is the limit as → 0 of
the space spanned by the following five matrices:
⎛0 02
⎜ ⎜
⎜1 ⎜
⎜0 0
⎝0 0
0
0
0
0
0
0
0
0
0
0
0⎞
0⎟
⎟
0⎟ ,
⎟
0⎟
0⎠
⎛0
⎜0
⎜
⎜0
⎜
⎜1
⎝0
0
0
0
0
0
0 0
0 0
0 0
0 2
0 0
0⎞
0⎟
⎟
0⎟ ,
⎟
0⎟
0⎠
⎛0
⎜0
⎜
⎜0
⎜
⎜0
⎝1
0
0
0
0
0
0
0
0
0
0
0 0 ⎞ ⎛0
0 0 ⎟ ⎜0
⎟ ⎜
0 0 ⎟ , ⎜1
⎟ ⎜
0 0 ⎟ ⎜0
0 2 ⎠ ⎝0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0⎞
0⎟
⎟
0⎟ ,
⎟
0⎟
0⎠
0 0
0 ⎞ ⎛0
0 0
0 ⎟ ⎜0
⎟ ⎜
0 0
0 ⎟ , ⎜1
⎟ ⎜
0 0
0 ⎟ ⎜0
0 −2 2 ⎠ ⎝0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
−5 6
⎛
4
−5
⎜ − ⎜ −2
⎜
− 2
⎜
⎜ −1 3 −4
⎝ −1 3 −4
0
0
0
0
0
0⎞
0⎟
⎟
0⎟ .
⎟
0⎟
0⎠
TN11,13 is the limit of the space spanned by
⎛0 02
⎜ ⎜
⎜1 ⎜
⎜0 0
⎝0 0
0
0
0
0
0
0
0
0
0
0
0⎞ ⎛ 0
0⎟ ⎜ 0
⎟ ⎜
0⎟ , ⎜−3
⎟ ⎜
0⎟ ⎜ 1
0⎠ ⎝ 0
0
0
0
0
0
0 0
0 0
0 1
0 3
0 0
0⎞ ⎛0
0⎟ ⎜0
⎟ ⎜
0⎟ , ⎜0
⎟ ⎜
0⎟ ⎜0
0⎠ ⎝1
0
0
0
0
0
0⎞ ⎛ 2 −5 6 −4
0⎟ ⎜ − 4 −5 3
⎟ ⎜
0⎟ , ⎜−2 − 2 −1
⎟ ⎜
0⎟ ⎜ −1 3 −4 2
0⎠ ⎝ −1 3 −4 2
TN14 is isomorphic to the Coppersmith-Winograd tensor.
We determine the rank of each tensor. In particular, we find two counterexamples to a
conjecture of J. Rhodes [3, Conjecture 0]. Special cases of the conjecture, mostly in small
dimension, were verified in [5, 3, 11]. The counterexamples are minimal, in the sense that as
we have shown above, a border rank 4 tensor in C4 ⊗C4 ⊗C4 can have rank at most 7, i.e. the
conjecture holds in this range.
Proposition 6.5.
R(TN1,4 ) = R(TN10,12 ) = R(TN11,13 ) = R(TN14 ) = R(TN15 ) = R(TN16 ) = R(TN1,4 ) = 9
and
R(TN6,8 ) = R(TN7,9 ) = 10.
Proof. The fact that rank of any tensor is at least 9 follows by the substitution method. To see
that R(TN6,8 ) ≥ 10, first apply Proposition 3.1 to the first and fourth row and to the third and
0⎞
0⎟
⎟
0⎟ .
⎟
0⎟
0⎠
22
J.M. LANDSBERG AND MATEUSZ MICHALEK
fifth column. This shows that the rank is at least 4 plus the rank of the tensor associated to
0
e ⎞
⎛ b
⎜c + αe b + βe d + γe⎟ ,
⎠
⎝ e
0
where α, β, γ are some constants. Now apply the proposition to the second column obtaining a
tensor represented by
b
e
⎛
⎞
⎜c + αe + δb d + γe + ρb⎟ ,
⎝
⎠
e
0
where δ, ρ, α, γ are (possibly new) constants. This space is equal to
⎛b e ⎞
⎜c d⎟
⎝e 0⎠
and it remains to show that it corresponds to a tensor of rank at least 5. This follows by
Proposition 3.1 by first reducing b, c, d and obtaining a rank 2 matrix.
To prove that R(TN7,9 ) ≥ 10, apply Proposition 3.1 and remove the second, third and fifth
column and first and fourth row to obtain a tensor isomorphic to
⎛ b d⎞
⎜c e⎟ ,
⎝0 b ⎠
and conclude as above.
The upper bounds for ranks of TN1,4 , TN10,12 , TN15 and TN17 follow from Proposition 6.3.
For TN6,8 , consider:
(1) 5 matrices, including the matrix corresponding to c, which follow from Proposition 6.3
for the upper left 3 × 3 corner,
(2) 2 matrices for last 2 diagonal entries,
(3) 1 matrix corresponding to d,
(4) the 2 matrices:
⎛0 0 0 0 0⎞ ⎛0 0 0 0 0⎞
⎜0 0 0 1 0⎟ ⎜0 0 0 0 0⎟
⎜
⎟ ⎜
⎟
⎜0 0 0 0 0⎟ , ⎜1 0 0 0 1⎟ .
⎜
⎟ ⎜
⎟
⎜0 0 0 0 0⎟ ⎜0 0 0 0 0⎟
⎝0 0 0 0 0⎠ ⎝1 0 0 0 1⎠
For TN11,13 it is enough to notice that once a matrix for c and the fourth diagonal entry
are given, one can generate the matrix corresponding to d using
⎛0
⎜0
⎜
⎜1
⎜
⎜1
⎝0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0⎞
0⎟
⎟
0⎟ .
⎟
0⎟
0⎠
An analogous method shows R(TN16 ) = 9. The tensor TN14 is a special case of Proposition
7.1.
For TN7,9 , consider 3 rank one matrices corresponding to entries of the first column,
2 rank one matrices corresponding to third and fourth diagonal entry and one matrix
ABELIAN TENSORS
23
corresponding to e. Apart from these six matrices we are left with the tensor represented
by
⎛a d 0 ⎞
⎜ b 0 d⎟ .
⎝0 b a⎠
This tensor is isomorphic to the symmetric tensor given by the monomial xyz which has
Waring rank 4, see, e.g., [30] (the upper bound dates back at least to [17]), and thus
tensor rank at most 4.
Apart from them there are two families of End-closed 5-dimensional subalgebras.
(1) The subspace spanned by identity
algebra
⎛0
⎜0
⎜
⎜0
⎜
⎜a
⎝d
and any 4-dimensional subspace of the 6-dimensional
0
0
0
b
e
0
0
0
c
f
0
0
0
0
0
0⎞
0⎟
⎟
0⎟ .
⎟
0⎟
0⎠
In this case there are normal forms: Tensors in A′ ⊗B ′ ⊗C ′ = C4 ⊗C2 ⊗C3 are classified
in [10]. We have T = a1 (b1 c1 +..+b5 c5 )+a2 b1 c4 +a3 b1 c5 +a4 b2 c4 +a5 b2 c5 +a6 b3 c4 +a7 b3 c5 =
a1 (b1 c1 + .. + b5 c5 ) + T ′ where a2 , ..a7 satisfy two linear relations. If we make a change of
basis in c4 , c5 , say by a 2 × 2 matrix X, then as long as we change b4 , b5 by X −1 the first
term does not change. Similarly, if we make a change of basis in b1 , b2 , b3 , by a matrix
Y , then as long as we change c1 , c2 , c3 by Y −1 , the first term does not change. In our
case we may assume the tensors are A-concise. There are the following cases (numbers
as in [10]), we abuse notation, writing T for T (A∗ ):
⎛x1
⎞
⎛x1
⎞
⎛x1
⎞
x1
x1
x1
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
x1
x1
x1
⎟ , T19 = ⎜
⎟ T20 = ⎜
⎟,
T9 = ⎜
⎜
⎟
⎜
⎟
⎜
⎟
⎜x2 x3
⎟
⎜x2 x3 x5 x1
⎟
⎜x2 x4 x5 x1
⎟
x1
⎝x4 x5
⎝x3 x4
⎝x3
x1 ⎠
x1 ⎠
x1 ⎠
⎞
⎛x1
⎞
⎛x1
⎞
⎛x1
x1
x1
x1
⎜
⎟
⎜
⎟
⎜
⎟
⎟
⎜
⎟
⎜
⎟
⎜
x1
x1
x1
⎟ , T23 = ⎜
⎟.
⎟ , T22 = ⎜
T21 = ⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎜x2 x4
⎟
⎜x2 x3 x4 x1
⎟
⎜x2 x4 x5 x1
⎟
x1
⎠
⎝
⎝x3 x5
⎠
⎝
x3 x4 x5
x1 ⎠
x1
x3
x5
x1
Now R(T9 ) = R(T20 ) = 5 because they are special cases of flag-algebra tensors - cf. Proposition 6.4.
T22 (A∗ ) is the limit of the space spanned by the following 5 matrices:
2
⎛
⎜
⎜
⎜
⎜
⎜
⎝1
−3
−
4 ⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟,⎜
⎟ ⎜
⎟ ⎜1 −
2⎠ ⎝
2
⎞ ⎛ 2
⎟ ⎜ ⎟ ⎜
⎟,⎜
⎟ ⎜
⎟ ⎜ ⎠ ⎝
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟,⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
2
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟,⎜
⎟ ⎜
⎟ ⎜1
⎠ ⎝1
⎞
⎟
⎟
⎟.
⎟
⎟
⎠
24
J.M. LANDSBERG AND MATEUSZ MICHALEK
T23 (A∗ ) is the limit of the space spanned by the following 5 matrices:
⎛
⎜
⎜
⎜ −26 4
⎜
⎜
⎝ −22 1
4
⎞ ⎛
⎟ ⎜−3
8⎟ ⎜
⎟,⎜
⎟ ⎜
⎟ ⎜ −2
4⎠ ⎝
− 12 8 ⎞ ⎛
3
4
5
1 7
⎟ ⎜ 2
2
⎟ ⎜
⎟,⎜
⎟ ⎜
⎟ ⎜ 1 22
4
⎠ ⎝ 2 23
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟,⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
⎞
⎟
⎟
⎟.
⎟
⎟
⎠
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟,⎜
⎟ ⎜
⎟ ⎜ 1 −
⎠ ⎝− 2
22
1 + 23
T21 (A∗ ) is the limit of the space spanned by the following 5 matrices:
⎛
⎜
⎜ 3 4
⎜ ⎜
⎜ 2
⎝ 2 3
4
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟,⎜
⎟ ⎜
⎟ ⎜1
⎠ ⎝
⎞ ⎛
4
⎟ ⎜ −
⎟ ⎜
3
⎟,⎜ ⎟ ⎜
⎟ ⎜ ⎠ ⎝
−6 8 ⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟,⎜
⎜
2
4 ⎟
− ⎟ ⎜
⎠ ⎝ −2 −3
⎞
⎟
⎟
⎟.
⎟
⎟
⎠
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟,⎜
⎟ ⎜
⎟ ⎜1
4⎠ ⎝
T19 (A∗ ) is the limit of the space spanned by the following 5 matrices:
2
⎛
⎜
⎜
⎜
⎜
⎜1
⎝
−3 4 ⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟,⎜
⎜
2 ⎟
− ⎟ ⎜
⎠ ⎝
2
⎞ ⎛
4 ⎟ ⎜
⎟ ⎜
⎟,⎜
⎟ ⎜
⎟ ⎜1 1
2 ⎠ ⎝1 1
⎞ ⎛ 2
⎟ ⎜ ⎟ ⎜
⎟,⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝ 1
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟,⎜
⎟ ⎜
⎟ ⎜−1 1
⎠ ⎝ 1 −1
⎞
⎟
⎟
⎟.
⎟
⎟
⎠
Proposition 6.6. R(T21 ) = 10 and all other tensors on this list have R(Tj ) = 9.
Proof. This follows by the substitution-method and considering the ranks of T ′ in [10].
(2) The subspace spanned by the identity and any 4-dimensional subspace of the 5-dimensional
algebra
(7)
⎛0
⎜a
⎜
⎜b
⎜
⎜0
⎝d
0
0
a
0
0
0
0
0
0
0
0
0
e
0
c
0⎞
0⎟
⎟
0⎟ .
⎟
0⎟
0⎠
All the operations we will perform preserve the identity matrix.
First, by exchanging rows and columns the algebra is
⎛0
⎜0
⎜
⎜a
⎜
⎜d
⎝b
0
0
0
c
e
0
0
0
0
a
0
0
0
0
0
0⎞
0⎟
⎟
0⎟ .
⎟
0⎟
0⎠
Note that the tensor TLeit,5 of Proposition 5.1 is in this family (set b = 0).
Proposition 6.7. All End-closed tensors in C5 ⊗C5 ⊗C5 obtained from 4-dimensional subspaces
of (7) have border rank five.
To prove the proposition, we will use the following lemma:
Lemma 6.8. Each of the tensors in C5 ⊗C5 ⊗C5 corresponding to linear spaces spanned by the
identity and the following subspaces have border rank 5:
ABELIAN TENSORS
⎛0
⎜0
⎜
S1 = ⎜a
⎜
⎜d
⎝b
0
0
0
0
e
0
0
0
0
a
0⎞
⎛0
0⎟
⎜0
⎜
⎟
0⎟ , S2 = ⎜a
⎜
⎟
⎜d
0⎟
⎝b
⎠
0
0
0
0
0
0
0
0
0
a
e
0
0
0
0
a
0
0
0
0
0
25
0⎞
⎛0
0⎟
⎜0
⎟
⎜
0⎟ , S3 = ⎜a
⎟
⎜
⎜d
0⎟
⎠
⎝b
0
0
0
0
e
0
0
0
0
0
a
0
0
0
0
0
0⎞
⎛0
0⎟
⎜0
⎟
⎜
0 ⎟ , S4 = ⎜a
⎟
⎜
⎜d
0⎟
⎠
⎝b
0
0
0
0
e
d
0
0
0
0
a
0
0
0
0
0
0⎞
0⎟
⎟
0⎟ .
⎟
0⎟
0⎠
The tensors corresponding to S1 , S3 , S4 , have rank 9 and the tensor corresponding to S2 has
rank 10.
Proof. We first prove the statement about the border rank. The span of S1 and the identity is
the limit of the space spanned by
⎛0
⎜0
⎜
⎜
⎜
⎜0
⎝2
0
0
0
0
0
0
0
1 2
2
0
0
0
0
0
0
0⎞ ⎛ 2
0⎟ ⎜ 0
⎟ ⎜
0⎟ , ⎜−
⎟ ⎜
0⎟ ⎜ 0
0⎠ ⎝ 2
⎛0 02
⎜0 ⎜
⎜0 0
⎜
⎜0 0
⎝0 0
0
0
0
0
0 − 12 3
0
0
1 2
0 2
0
0
0 −
0
0
0
0
0
0 12 4 ⎞ ⎛0
0
0
0 ⎟ ⎜
⎜
1 3 ⎟ ⎜0
⎟
0 −2 ⎟ , ⎜
0
0 ⎟ ⎜
⎜
0
2 ⎠ ⎝0
0⎞ ⎛0
0
0⎟ ⎜
⎟ ⎜
⎜
0⎟ , ⎜0
⎟
0⎟ ⎜
⎜
0⎠ ⎝4
0
0
0
0
0
0
0
0
2
4
0 0
0 0
0 0
0 2
0 0
0⎞
0⎟
⎟
0⎟
⎟,
0⎟
⎟
0⎠
0⎞
⎛ 0
0⎟
⎜ 0
⎟
⎜
0⎟ , X3 = ⎜ 0
⎟
⎜
⎜ 0⎟
⎠
⎝1 + α
0
0
0
0
0
0
0
0
0
0
0
0
2
4
0
0⎞
0⎟
⎟
0⎟
⎟.
0⎟
⎟
0⎠
For S2 , consider
⎛
⎜0
⎜
X1 = ⎜2
⎜
⎜0
⎝1
4
0 6
0 0
0 4
0 0
0 2
⎛0
⎜0
⎜
X4 = ⎜
⎜0
⎜0
⎜
⎝0
0
0
0
0
0
0 8 ⎞
⎛ 03
4
0 0⎟
αβ
β
⎜
⎟
⎜
0
0 6 ⎟ , X2 = ⎜ 0
⎟
⎜
⎜ 0 0⎟
α−1 2
⎝ α
0 4 ⎠
0
0
0
0
0
0
0
0
α
2
3
α
β(α−1)
0⎞
⎛ 0
⎜ α3
0⎟
⎟
⎜
2
⎜
0⎟
⎟ , X5 = ⎜ α
⎟
⎜
0⎟
⎜ β(α−1) 0⎠
⎝ 0
0
0
0 αβ 2 6
0
0
0 β4
0 αβ3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1−α 2
α 0
0
0
0
0
0
0
0
0
0
0⎞
0⎟
0⎟
⎟,
⎟
0⎟
0⎠
0⎞
0⎟
⎟
0⎟
⎟,
0⎟
⎟
0⎠
where in order that X3 has rank one, take α to be a solution of the equation α2 + α − 1 = 0. We
determine β later.
First note that the limits as goes to zero of
1
1
X5 , X1 , X4
give matrices corresponding respectively to d, b, e.
Consider X1 + X2 − X3 . The constant terms and the terms of order add to zero. The (4, 2)
entry equals
1 1−α 2 2
( −
) = .
α
α
Hence, the limit gives the matrix corresponding to a.
26
J.M. LANDSBERG AND MATEUSZ MICHALEK
It remains to prove that the identity matrix belongs to the limit. For this we consider
X1 +
1
1
(X2 −
X3 ) − X4 − X5 .
β
1−α
As 4 is on the diagonal it remains to prove that the lower order terms all add to zero. For the
constant term
α
1+α
β(1 − α) + α − α2 − 1 − α
1+ −
=
β β(1 − α)
β(1 − α)
The numerator equals β(1 − α) − α2 − 1, so we take β =
terms proportional to cancel, observe that
α2 +1
1−α
to make it zero. To see that the
1
1
α
(1 −
)=−
.
β
1−α
β(α − 1)
All three of the terms proportional to 2 cancel, the only nontrivial being
1
1 1−α
−
.
α 1−α α
The term proportional to 3 also cancels out.
The span of S3 and the identity is the limit of the space spanned by
⎛0
⎜0
⎜
⎜−
⎜
⎜0
⎝2
0
0
0
0
0
0
0
1 2
2
0
−
0
0
0
0
0
0⎞ ⎛2
0⎟ ⎜ 0
⎟ ⎜
0⎟ , ⎜ ⎟ ⎜
0⎟ ⎜ 0
0⎠ ⎝ 2
⎛0 02
⎜0 ⎜
⎜0 0
⎜
⎜0 ⎝0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1 3
2
1 4
2 ⎞
0
0
0
0
0
0
1 2
2
0
0⎞ ⎛0
0 ⎟ ⎜0
⎟ ⎜
0⎟ , ⎜0
⎟ ⎜
0 ⎟ ⎜1
0⎠ ⎝4
⎛0 0 0 0
0 ⎟ ⎜0 0 0 0
1 3 ⎟ ⎜0
0 0 0
⎟,⎜
2 ⎟ ⎜
0 ⎟ ⎜1 − 0 2
2 ⎠ ⎝0 0 0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0⎞
0⎟
⎟
0⎟ ,
⎟
0⎟
0⎠
0⎞
0⎟
⎟
0⎟ .
⎟
0⎟
0⎠
The span of S4 and the identity is the limit of the space spanned by
⎛0
⎜0
⎜
⎜0
⎜
⎜0
⎝1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
⎛0 06
⎜0 ⎜
⎜0 0
⎜
⎜0 −4
⎝0 0
0⎞ ⎛6 8
0⎟ ⎜ 0 0
⎟ ⎜
0⎟ , ⎜ 0 0
⎟ ⎜
0⎟ ⎜2 4
0⎠ ⎝ 1 2
0 0
0 −8
0 0
0 6
0 0
0
0
0
0
0
0 12 ⎞ ⎛ 0
0 0 ⎟ ⎜0
⎟ ⎜
0 0 ⎟ , ⎜3
⎟ ⎜
0 8 ⎟ ⎜ 0
0 6 ⎠ ⎝ 1
0
0
0⎞ ⎛ 0
0
0
0⎟ ⎜ 0
⎟ ⎜
9
0⎟ , ⎜−3 8
⎟ ⎜
8
0⎟ ⎜−2 7
−3
2
⎠
⎝
0
− −3
0 0
0 0
0 6
0 0
0 3
0
0
0
0
0
0
0
0
0
0
0⎞
0⎟
⎟
0⎟ ,
⎟
0⎟
0⎠
0⎞
0⎟
⎟
0⎟ .
⎟
0⎟
0⎠
The lower bounds for rank follow by substitution method. For rank upper bounds, consider
the seven rank one matrices:
(1) 4 matrices corresponding to first two and last two entries of the diagonal,
(2) 1 matrix corresponding to b,
ABELIAN TENSORS
27
(3) the 2 matrices:
⎛0
⎜0
⎜
⎜−1
⎜
⎜0
⎝1
0 0
0 0
0 1
0 0
0 −1
0
0
0
0
0
0⎞ ⎛0
0⎟ ⎜0
⎟ ⎜
0⎟ , ⎜1
⎟ ⎜
0⎟ ⎜0
0⎠ ⎝1
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0⎞
0⎟
⎟
0⎟ .
⎟
0⎟
0⎠
In all four cases it is easy to find the remaining rank 1 matrices.
Proof of Proposition 6.7. If the subspace is given by a = 0 then we conclude by Proposition
6.4. Otherwise there exists a matrix M1 in the algebra with the entries corresponding to a
nonzero. We may assume that the 3 other generators of the algebra M2 , M3 , M4 have entries
corresponding to a equal to zero. Further, we may assume that M2 has only one entry nonzero,
corresponding to b, as otherwise, by considering M12 the algebra would not be End-closed. Hence
we may assume that M3 and M4 have only nonzero entries on d, c and e.
Let S̃ denote the 3-dimensional vector space corresponding to d, c and e, and let S ⊂ S̃ be
the two-dimensional subspace spanned by M3 and M4 .
Case 1) S = {M ∈ S̃ ∣ c = 0}. We may assume that M3 corresponds to e, M4 corresponds to d
and the algebra is given by
⎛0 0 0
⎜0 0 0
⎜
⎜a 0 0
⎜
⎜d λa 0
⎝b e a
0
0
0
0
0
0⎞
0⎟
⎟
0⎟ ,
⎟
0⎟
0⎠
for some constant λ. If λ = 0 we are in case S1 . Otherwise, by multiplying second column by
1/λ and second row by λ we may assume that λ = 1 and are in case S2 .
Case 2) S = {M ∈ S̃ ∣ e = 0}. Subtract any multiple of the third column from the second
column, and add the same multiplicity of the second row to the third row to reduce to case S3 .
Case 3) S = {M ∈ S̃ ∣ d = 0}. This is analogous to Case 2).
Case 4) As we are not in case 2) or 3) we may assume there are constants λ ∈ C, and δ ≠ 0
such that
0
⎛0
0
⎜0
⎜
0
S = ⎜0
⎜
⎜d λd + δe
⎝0
e
0
0
0
0
0
0
0
0
0
0
Then we may assume that M1 represents the space
⎛0 0
⎜0 0
⎜
⎜a 0
⎜
⎜0 0
⎝0 ρa
0
0
0
0
a
0
0
0
0
0
0⎞
0⎟
⎟
0⎟ .
⎟
0⎟
0⎠
0⎞
0⎟
⎟
0⎟ .
⎟
0⎟
0⎠
28
J.M. LANDSBERG AND MATEUSZ MICHALEK
Subtract ρ times the third column from the second column, and add ρ times the second row to
third row, reducing to ρ = 0. At this point we have the algebra
0
⎛0
0
⎜0
⎜
0
⎜a
⎜
⎜d λd + δe
⎝b
e
0
0
0
0
a
0
0
0
0
0
0⎞
0⎟
⎟
0⎟ .
⎟
0⎟
0⎠
Subtract 1/δ times the fourth row from the fifth row and add 1/δ times the fifth column to the
fourth column to obtain a subspace isomorphic to
0
⎛0
0
0
⎜
⎜a
0
⎜
⎜
⎜d
e
⎝b − λ d
δ
0
0
0
0
a
0
0
0
0
0
0⎞
0⎟
0⎟
⎟.
⎟
0⎟
0⎠
If λ = 0 we are in Case 2). If λ ≠ 0 we multiply the second column by − λδ and the second row by
− λδ to reduce to case S4 .
6.5. Examples with large gaps between rank and border rank. First consider
Tgap = a1 ⊗(b1 ⊗c1 +⋯+b6 ⊗c6 )+a2 ⊗b6 ⊗c1 +a3 ⊗(b5 ⊗c1 +b6 ⊗c3 )+a4 ⊗b5 ⊗c2 +a5 ⊗b4 ⊗c3 +a6 ⊗(b4 ⊗c2 +b5 ⊗c3 ),
i.e.,
⎛x1
x1
⎜
⎜
⎜
∗
Tgap (A ) = ⎜
⎜
x6
⎜
⎜x3 x4
⎝x
2
x1
x5 x1
x6
x1
x3
⎞
⎟
⎟
⎟
⎟.
⎟
⎟
⎟
x1 ⎠
Note Tgap is neither 1B nor 1C -generic. In the following proposition we show that Tgap is also a
counterexample to [3, Conjecture 0].
Proposition 6.9. R(Tgap ) = 6 and R(Tgap ) = 12.
Proof. To prove that R(Tgap ) = 12, by the substitution method, it suffices to prove that R(T ′ ) =
6, where T ′ corresponds to the subspace
x6 x5 ⎞
⎛
⎜x3 x4 x6 ⎟ .
⎝x2
x3 ⎠
This space is contained in the space spanned by the following 6 rank one matrices:
(1) 3 matrices corresponding to x2 , x4 , x5 ,
(2) 2 matrices corresponding to x6 ,
(3) the matrix
⎛
⎜1
⎝1
⎞
1⎟ .
1⎠
ABELIAN TENSORS
29
Were T ′ of rank 5, then the space would be equal to the space generated by five rank one
matrices. However, each rank one matrix in this space has x3 = 0. This finishes the proof
that R(T ) = 12. It remains to prove R(Tgap ) = 6. As Tgap is A-concise we only need to prove
R(Tgap ) ≤ 6. Consider the following 6 rank one matrices:
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝1
5
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜4
⎜
⎝1
9
8
4
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟,⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟,⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎜−4 −1
⎠ ⎝
1
10 ⎞ ⎛
⎟ ⎜ 5 8
⎟ ⎜
⎟ ⎜
⎟,⎜ 3 6
⎟ ⎜ ⎟ ⎜
9⎟ ⎜
⎟ ⎜ 1 3
5 ⎠ ⎝
⎞
⎟
⎟
⎟
⎟,
⎟
⎟
⎟
⎟
⎠
⎞ ⎛
⎜
10 ⎟
⎟ ⎜
⎟ ⎜ 8
5 −10
⎟
⎜
,
3
⎜
−1
5
8 ⎟
⎟ ⎜ −
⎟
⎜
6
3
5
8
⎟ ⎜ − −
7
4
⎠ ⎝ − −
9
⎞
⎟
⎟
⎟
⎟.
⎟
⎟
⎟
⎟
⎠
So far all the tensors we considered had rank less than or equal to twice the border rank. By
[7] the maximal rank of a tensor is at most twice the maximal border rank, and all previously
known examples of tensors had rank at most twice the border rank. The following example goes
beyond this ratio.
For T ∈ A⊗B⊗C and T ′ ∈ A′ ⊗B ′ ⊗C ′ , consider A⊗A′ as a single vector space and similarly
for B⊗B ′ and C⊗C ′ . Define T ⊗T ′ ∈ (A⊗A′ )⊗(B⊗B ′ )⊗(C⊗C ′ ). In what follows, we will use a
tensor T ′ ∈ C2 ⊗C2 ⊗C2 to produce three copies of Tgap .
Proposition 6.10. The tensor
Tbiggap ∶= Tgap ⊗ (e1 ⊗ f1 ⊗ g1 + e2 ⊗ (f2 ⊗ g1 + f1 ⊗ g2 )) ∈ (C12 )⊗3
has border rank 12 and rank at least 25.
Proof. Tbiggap has border rank 12 as it is the tensor product of concise tensors of border rank 6
and 2. In terms of matrices,
Tbiggap (A∗ ⊗C2∗ ) = (
Tgap (A∗ ) Tgap (Ã∗ )
)
Tgap (Ã∗ )
where Ã∗ is another copy of A∗ . Denote the vectors in T (Ã∗ ) with primes. Eliminate x1 , x′1 by
the substitution method so that Tbiggap has border rank at least 9 plus the rank of the tensor
T̃ ∈ C6 ⊗C6 ⊗C10 represented by
⎛0
⎜x3
⎜x
⎜
M ∶= ⎜ 2
⎜0
⎜ ′
⎜x3
⎝x′
2
x6
x4
0
x′6
x′4
0
x5 0 x′6 x′5 ⎞
x6 x′3 x′4 x′6 ⎟
x3 x′2 0 x′3 ⎟
⎟
⎟.
x′5 0 0 0 ⎟
⎟
x′6 0 0 0 ⎟
x′3 0 0 0 ⎠
We now prove R(T̃ ) ≥ 13. Write T̃ = x2 ⊗M2 +⋯+x′6 ⊗M6′ . Apply Proposition 3.1 first with x2 to
get a tensor T̃ (1) = x3 ⊗(M3 −λ3 M2 )+⋯+x′6 ⊗(M6′ −λ′6 M2 ) with R(T̃ (1) ) ≤ R(T̃ )−1. Continue in
30
J.M. LANDSBERG AND MATEUSZ MICHALEK
this manner, eliminating all but x′3 to get a tensor T̃ (9) = x3 ⊗(M3 +c2 M2 +⋯+c′6 M6′ ) ∈ C1 ⊗C6 ⊗C6
where the cj , c′j are some constants.
Hence R(T̃ ) ≥ 9 + R(T̃ (9) ). But R(T̃ (9) ) is simply the (usual) rank of the matrix
⎛0
⎜c3
⎜c
⎜
M̃3 = ⎜ 2
⎜0
⎜
⎜1
⎝c′
2
c6
c4
0
c′6
c′4
0
c5 0 c′6 c′5 ⎞
c6 1 c′4 c′6 ⎟
c3 c′2 0 1 ⎟
⎟
⎟.
c′5 0 0 0 ⎟
⎟
c′6 0 0 0 ⎟
1 0 0 0⎠
It remains to show that rank(M̃3 ) ≥ 4. Suppose to the contrary that all the 4 × 4 minors of M̃3
are zero. One of the minors equals (1 − c′2 c′6 )2 , hence we would need c′2 , c′6 ≠ 0. There is also a
2
minor (c′2 c′5 )2 , which would force c′5 = 0. However, there is also the minor (c′4 c′5 − c′2
6 ) which
under these assumptions cannot be zero. We conclude R(T ) ≥ 12 + 9 + 4 = 25.
By further tensoring Tgap analogously as above, we obtain tensors with rank to border rank
ratio converging at least to 13/6.
The following tensor is a generalization of S2 of Lemma 6.8, which is the case Tbiggap,5 .
Theorem 6.11. Let m = 2k + 1. Let T̃ ∈ Cm−1 ⊗Ck+1 ⊗Ck+1 be the tensor represented by
0
0
⎛ x0
x0
0
⎜ x1
⎜ x
0
x0
⎜ 2
⎜
⎜ ⋮
⎜
⎜xk−1
0
0
⎝ x
xk+1 xk+2
k
...
...
⋮
0
0
0
...
...
x0
x2k−1
0⎞
0⎟
0⎟
⎟
⎟.
⎟
⎟
0⎟
x0 ⎠
Let Tbiggap,m ∈ Cm ⊗Cm ⊗Cm be the tensor such that Tbiggap,m (A∗ ) has T̃ (Cm−1∗ ) in its lower
left (k + 1) × (k + 1) corner and x2k along the diagonal. Then m ≤ R(Tbiggap,m ) ≤ m + 1 and
R(Tbiggap,m ) = 25 m. In particular the ratio of rank to border rank tends to 52 as m → ∞.
Proof. It is straightforward that R(T̃ ) = 3k. By the substitution method R(Tbiggap,m ) ≥ 2k +
R(T̃ ) = 5k and in fact equality holds.
To estimate the border rank, fix bases {aj } of A, {bj } of B and {cj } of C so that T̃ belongs to
the subspace span{a1 , . . . , am−1 }⊗span{bk+1 , . . . , bm }⊗span{e1 , . . . , ek+1 }. Consider the following
m rank 1 elements of B⊗C:
(1) (2bm + bk+1 + 2 b1 )⊗(c1 + 21 ck+1 + 21 2 cm ),
(2) (2bm − bk+1 + 2 b1 )⊗(c1 − 21 ck+1 + 21 2 cm ),
(3) (bk+1 + ⋅ ⋅ ⋅ + bm−1 + 2bm )⊗c1 ,
(4) bm ⊗(2c1 + c2 + ⋅ ⋅ ⋅ + ck ),
(5) bi ⊗(c1 + ci−k+1 + 2 ci ) for i = k + 2, . . . , m − 1,
(6) (bm − bk+i + 2 bi )⊗ci for i = 2, . . . , k.
The rank one elements of T̃ (Cm−1∗ ) are obtained from 5) and 6). The diagonal of T̃ (Cm−1∗ )
is obtained by adding all elements of 5) with 1) and subtracting 3). The identity matrix is
obtained by adding all elements of 5) and 6) with 1) and 2) and subtracting 3) and 4).
Remark 6.12. After we posted a preprint of this article on the arXiv, Jeroen Zuiddam shared
with us his forthcoming article presenting an example of a sequence of tensors with rank to
border rank ratio approaching three.
ABELIAN TENSORS
31
Question 6.13. Is the ratio of rank to border rank unbounded? Can one find explicit tensors
with ratio 3 or larger?
In this context we recall the following problem, which is a variant of our question in the
situation of minimal border rank:
Problem 6.14. [6, Open problem 4.1] Is there an explicit family of tensors Tm ∈ Cm ⊗Cm ⊗Cm
with R(Tm ) ≥ (3 + )m for some > 0? Can we even achieve this for tensors corresponding to
the multiplication in an algebra, i.e., is there an explicit family of algebras Am with R(Am ) ≥
(3 + ) dim Am for some > 0?
For tensors T ∈ A1 ⊗⋯⊗ An , there are tensors of rank n − 1 of border rank two.
6.6. There are parameters worth of non-isomorphic 1-generic border rank m tensors
in Cm ⊗Cm ⊗Cm . Let τ ∈ M atp×n . Set m = n + p + 1. Define
p
n
n
j=1
s=1
s=1
TLeit,τ ∶= a1 ⊗(b1 ⊗c1 + ⋯⊗bp+n+1 ⊗cp+n+1 ) + ∑ a1+j ⊗bj ⊗( ∑ τj,s cp+1+s ) + ∑ ap+1+s bp+1 cp+1+s
Leitner [33] shows that R(TLiet,τ ) = m and that the family gives non-isomorphic tensors for
p ≥ 4, n ≥ 2.
Remark 6.15. Leitner only shows the border rank condition under certain genericity hypotheses
on τ , but from the border rank perspective they are unnecessary by taking limits. (Border rank
is semi-continuous.)
In particular, when p = 4, n = 2 Lietner shows that there is a one-parameter family of nonisomorphic subgroups. The same argument shows that there is a corresponding one-parameter
family of non-isomorphic tensors.
7. Coppersmith-Winograd value
As mentioned in the introduction, a motivation for this article is the study of upper bounds for
the exponent of matrix multiplication. For our purposes, the exponent ω of matrix multiplication,
2
2
2
which governs the complexity of the matrix multiplication tensor M⟨n⟩ ∈ Cn ⊗Cn ⊗Cn , may be
defined as
ω ∶= inf{τ ∈ R ∣ R(M⟨n⟩ ) = O(nτ )}.
Naı̈vely one has ω ≤ 3 and it is generally conjectured by computer scientists that ω = 2.
The “proper” way to determine the exponent of matrix multiplication would be to determine
the border rank of the matrix multiplication tensor. Unfortunately, this appears to be beyond
our current capabilities. Thanks to a considerable amount of work, most notably [37, 40, 15],
we can prove upper bounds for matrix multiplication by considering other tensors.
First, Schönhage’s asymptotic sum inequality [37] states that for all li , mi , ni , with 1 ≤ i ≤ s,
s
ω
s
∑(mi ni li ) 3 ≤ R(⊕ M⟨mi ,ni ,li ⟩ ).
i=1
i=1
Then Strassen [40] pointed out that it would be sufficient to find upper bounds on the border
rank of a tensor that degenerated into a disjoint sum of matrix multiplication tensors. This was
exploited most successfully by Coppersmith and Winograd [15], who attained their success with
a tensor T̃CW . The purpose of this section is to isolate geometric aspects of this tensor in the
hope of finding other tensors that would enable further upper bounds on the exponent.
In practice, only tensors of minimal, or near minimal border rank have been used to prove
upper bounds on the exponent. Call a tensor that gives a “good” upper bound for the exponent
32
J.M. LANDSBERG AND MATEUSZ MICHALEK
via the methods of [40, 15], of high Coppersmith-Winograd value or high CW-value for short. We
briefly review tensors that have been utilized. Our study is incomplete because the CW-value of
a tensor also depends on its presentation, and in different bases a tensor can have quite different
CW-values. Moreover, even determining the value in a given presentation still involves some
“art” in the choice of a good decomposition, choosing the correct tensor power, estimating the
value and probability of each block [43].
7.1. Schönhage’s tensors. Schönhage’s tensors are TSch = M⟨N,1,1⟩ ⊗M⟨1,m,n⟩ where N = (m −
1)(n−1). Here R(TSch ) = N +1 while R(TSch ) = N +mn = 2R(TSch )−(m+n−1). It gives ω < 2.55.
There is nothing to gain by taking tensor powers here because the two matrix multiplications
are already disjoint.
7.2. The Coppersmith-Winograd tensors. Coppersmith and Winograd define two tensors:
q
Tq,CW ∶= ∑ a0 ⊗bj ⊗cj + aj ⊗b0 ⊗cj + aj ⊗bj ⊗c0 ∈ Cq+1 ⊗Cq+1 ⊗Cq+1
(8)
j=1
and
(9)
q
T̃q,CW ∶= ∑ (a0 ⊗bj ⊗cj +aj ⊗b0 ⊗cj +aj ⊗bj ⊗c0 )+a0 ⊗b0 ⊗cq+1 +a0 ⊗bq+1 ⊗c0 +aq+1 ⊗b0 ⊗c0 ∈ Cq+2 ⊗Cq+2 ⊗Cq+2
j=1
both of which have border rank q + 2.
In terms of matrices,
xq ⎞
⎛ 0 x1 ⋯
⎜x1 x0 0 ⋯
⎟
⎜
⎟
⎟
Tq,CW (C ∗ ) = ⎜x2 0 x0
⎜
⎟
⎜⋮
⎟
⋮
⋱
⎝xq 0 ⋯ 0 x0 ⎠
and
xq x0 ⎞
⎛xq+1 x1 ⋯
x
x
0
⋯
0⎟
0
⎜ 1
⎜ x
⎟
0
x
⎜
⎟
0
⎟.
T̃q,CW (C ∗ ) = ⎜ 2
⎜ ⋮
⎟
⋮
⋱
⎜
⎟
⎜ xq
⎟
0 ⋯ 0 x0
⎝ x
0 ⋯
0 0⎠
0
Permuting bases, we may also write
⎛ x0
⎜ x1
⎜ x
⎜
T̃CW (C ∗ ) = ⎜ 2
⎜ ⋮
⎜
⎜ xq
⎝x
q+1
⎞
x0 0 ⋯
0⎟
⎟
0 x0
⎟
⎟.
⎟
⋮
⋱
⎟
⎟
0 ⋯ 0 x0
x1 ⋯
xq x0 ⎠
Proposition 7.1. R(Tq,CW ) = 2q + 1, R(T̃q,CW ) = 2q + 3.
Proof. We first prove the lower bound for Tq,CW . Apply Proposition 3.1 to show that the rank
of the tensor is at least 2q − 2 plus the rank of
0 x0
(
),
x0 x1
ABELIAN TENSORS
33
which has rank 3. An analogous estimate provides the lower bound for R(T̃q,CW ). To show that
R(Tq,CW ) ≤ 2q + 1 consider the following rank 1 matrices, whose span contains T (A∗ ):
1) q + 1 matrices with all entries equal to 0 apart from one entry on the diagonal equal to 1,
2) q matrices indexed by 1 ≤ j ≤ q, with all entries equal to zero apart from the four entries
(0, 0), (0, j), (j, 0), (j, j) equal to 1.
For the tensor T̃CW we consider the same matrices, however both groups have one more
element.
Coppersmith and Winograd used T̃CW to show ω < 2.3755. In subsequent work Stothers [38],
⊗4
⊗8
⊗16
⊗32
resp. V. Williams [43], resp. LeGall [32] used T̃CW
resp. T̃CW
, resp. T̃CW
and T̃CW
leading to
the current “world record” ω < 2.3728639.
Ambainis, Filmus and LeGall [2] showed that taking higher powers of T̃CW when q ≥ 5 cannot
prove ω < 2.30 by this method alone. Their suggestion that one should look for new tensors to
prove further upper bounds was one motivation for this paper.
7.3. Strassen’s tensor. Strassen uses the following tensor to show ω < 2.48:
q
(10)
TStr,q = ∑ a0 ⊗bj ⊗cj + aj ⊗b0 ⊗cj ∈ Cq+1 ⊗Cq+1 ⊗Cq
j=1
which has border rank q+1, as these are tangent vectors to q points of the Pq−1 = P{a0 ⊗b0 ⊗⟨c1 , ⋯, cq ⟩}
that lies on the Segre and TStr,q is concise. Note that it is a specialization of TCW obtained by
setting c0 = 0. By the substitution method the rank of the tensor equals 2q.
The corresponding linear spaces are:
⎛ 0 x1 ⋯ xq ⎞
⎜x1 0 ⋯ 0 ⎟
⎜
⎟
⎟,
TStr,q (C ∗ ) = ⎜x2
⎜
⎟
⎜⋮
⋮
⋮⎟
⎝ xq 0 ⋯ 0 ⎠
and
⎛x1 x2 ⋯ xq ⎞
⎜x0 0 ⋯ 0 ⎟
⎜
⎟
∗
TStr,q (A ) = ⎜ 0 x0 0 ⋮ ⎟ .
⎜
⎟
⎜⋮ ⋱
⎟
⎝ 0 ⋯ 0 x0 ⎠
Actually Strassen uses the tensor product of this tensor with its images under Z3 acting on
the three factors: T̃ ∶= T ⊗T ′ ⊗T ′′ where T ′ , T ′′ are cyclic permutations of T = TStr,q . Thus
2
2
2
T̃ ∈ Cq(q+1) ⊗Cq(q+1) ⊗Cq(q+1) and R(T̃ ) ≤ (q + 1)3 .
7.4. Extremal tensors. Let A, B, C = Cm . There are normal forms for germs of curves in
Seg(PA × PB × PC) up to order m − 1, namely
Tt = (a1 +ta2 +⋯+tm−1 am +O(tm ))⊗(b1 +tb2 +⋯+tm−1 bm +O(tm ))⊗(c1 +tc2 +⋯+tm−1 cm +O(tm ))
and if the aj , bj , cj are each linearly independent sets of vectors, we will call the curve general
to order m − 1.
Proposition 7.2. Let T ∈ A⊗B⊗C = Cm ⊗Cm ⊗Cm . If
(m−1)
T (A∗ ) = T0
(A∗ ),
with Tt a curve that is general to order m, then T (A∗ ) is the centralizer of a regular nilpotent
element.
34
J.M. LANDSBERG AND MATEUSZ MICHALEK
(q)
Proof. Note that T0
= q! ∑i+j+k=q−3 ai ⊗bj ⊗ck , i.e.,
⎛xq−2 xq−3
⎜xq−3 xq−4
⎜
⎜ ⋮
⎜
(q)
∗
0
T0 (A ) = ⎜
⎜ x1
⎜ 0
0
⎜
⎜
⎜ ⋮
⋮
⎝ 0
0
⋯ ⋯ x1 0 ⋯ ⎞
⋯ x1 0 ⋯ ⋯ ⎟
⎟
⎟
⎟
⎟
⋯
⎟
⎟
⋯
⎟
⎟
⎟
⎠
⋯
in particular, each space contains the previous ones, and the last equals
x1 ⎞
⎛ xm xm−1 ⋯
x
x
⋯
x
0⎟
⎜ m−1
m−2
1
⎜
⎟
⋮
⋰
⎜ ⋮
⎟
⎜
⎟
⎜ ⋮
⎟
x1
⎝ x1
⎠
0
which is isomorphic to the centralizer of a regular nilpotent element.
This provides another, explicit proof that the centralizer of a regular nilpotent element belongs
to the closure of diagonalizable algebras.
Theorem 7.3. Let T ∈ A⊗B⊗C = Cm ⊗Cm ⊗Cm be of border rank m > 2. Assume PT (A∗ ) ∩
Seg(PB × PC) = [X] is a single point, and PT̂[X] Seg(PB × PC) ⊃ PT (A∗ ). Then T is not
1A -generic.
If PT (A∗ )∩Seg(PB ×PC) = [X] is a single point, and PT̂[X] Seg(PB ×PC)∩PT (A∗ ) is a Pm−2
and T is 1A -generic, then T = T̃m−2,CW is isomorphic to the Coppersmith-Winograd tensor.
Proof. For the first assertion, no element of PT̂[X] Seg(PB × PC) has rank greater than two.
For the second, we first show that T is 1-generic. If we choose bases such that X = b1 ⊗c1 ,
then, after changing bases, the Pm−2 must be the projectivization of
(11)
x2 ⋯ xm−1 0⎞
⎛ x1
⎜ x2
⎟
⎜
⎟
⎟.
E ∶= ⎜ ⋮
⎜
⎟
⎜xm−1
⎟
⎝ 0
⎠
Write T (A∗ ) = span{E, M } for some matrix M . As T is 1A -generic we can assume that M
is invertible. In particular, the last row of M must contain a nonzero entry. In the basis order
x1 , . . . , xm−1 , M , the space of matrices T (B ∗ ) has triangular form and contains matrices with
nonzero diagonal entries. The proof for T (C ∗ ) is analogous, hence T is 1-generic.
By Proposition 5.8 we may assume that T (A∗ ) is contained in the space of symmetric matrices.
Hence, we may assume that E is as above and M is a symmetric matrix. By further changing
the basis we may assume that M has:
(1) the first row and column equal to zero, apart from their last entries that are nonzero (we
may assume they are equal to 1),
(2) the last row and column equal to zero apart from their first entries.
Hence the matrix M is determined by a submatrix M ′ of rows and columns 2 to m − 1. As
T (A∗ ) contains a matrix of maximal rank, the matrix M ′ must have rank m − 2. We can change
the basis x2 , . . . , xm−1 in such a way that the quadric corresponding to M ′ equals x22 + ⋅ ⋅ ⋅ + x2m−1 .
ABELIAN TENSORS
35
This will also change the other matrices, which correspond to quadrics x1 xi for 1 ≤ i ≤ m − 1,
but will not change the space that they span. We obtain the tensor T̃m−2,CW .
7.5. Compression extremality. Compression genericity is defined and discussed in [31]. Here
we just discuss the simplest case. We say a 1-generic, tensor T ∈ A⊗B⊗C is maximally compressible if there exists hyperplanes HA ⊂ A∗ , HB ⊂ B ∗ , HC ⊂ C ∗ such that T ∣HA ×HB ×HC = 0.
If T ∈ S 3 A ⊂ A⊗A⊗A, we will say T is maximally symmetric compressible if there exists a
hyperplane HA ⊂ A∗ such that T ∣HA ×HA ×HA = 0.
Recall that a tensor T ∈ Cm ⊗Cm ⊗Cm that is 1-generic and satisfies Strassen’s equations is
strictly isomorphic to a tensor in S 3 Cm .
Theorem 7.4. Let T ∈ S 3 Cm be 1-generic and maximally symmetric compressible. Then T is
one of:
(1) Tm−1,CW
(2) T̃m−2,CW
(3) T = a1 (a21 + ⋯a2m ). As a subspace of Cm ⊗Cm , this is
⋯ xm ⎞
⎛ x1 x2
⎜ x2 x1 0 ⋯
⎟
⎜
⎟
⎜ x3 0 x1
⎟.
⎜
⎟
⎜ ⋮
⎟
0
⋱
⎝xm 0
x1 ⎠
In particular, the only 1-generic, maximally symmetric compressible, minimal border rank tensor
in Cm ⊗Cm ⊗Cm is T̃m−2,CW .
Proof. Let a1 be a basis of the line HA ⊥ ⊂ Cm . Then T = a1 Q for some Q ∈ S 2 Cm . By 1genericity, the rank of Q is either m or m − 1. If the rank is m, there are two cases, either the
hyperplane HA is tangent to Q, or it intersects it transversely. The second is case 3. The first
has a normal form a1 (a1 am + a22 + ⋯ + a2m−1 ), which, when written as a tensor, is T̃m−2,CW . If Q
has rank m − 1, by 1-genericity, its vertex must be in HA and thus we may choose coordinates
such that Q = (a22 + ⋯ + a2m ), but then T , written as a tensor is Tm−1,CW .
References
1. Boris Alexeev, Michael Forbes, and Jacob Tsimerman, Tensor rank: some lower and upper bounds, IEEE
Conference on Computational Complexity, IEEE Computer Society, Feb 2011, pp. 283–291.
2. Andris Ambainis, Yuval Filmus, and François Le Gall, Fast matrix multiplication: Limitations of the laser
method, CoRR abs/1411.5414 (2014).
3. Edoardo Ballico and Alessandra Bernardi, Stratification of the fourth secant variety of Veronese varieties via
the symmetric rank, Adv. Pure Appl. Math. 4 (2013), no. 2, 215–250. MR 3069955
4. Daniel J. Bates and Luke Oeding, Toward a salmon conjecture, Exp. Math. 20 (2011), no. 3, 358–370.
MR 2836258 (2012i:14056)
5. Alessandra Bernardi, Alessandro Gimigliano, and Monica Idà, Computing symmetric rank for symmetric
tensors, J. Symbolic Comput. 46 (2011), no. 1, 34–53. MR 2736357 (2012h:14126)
6. Markus Bläser, Explicit tensors, Perspectives in Computational Complexity, Springer, 2014, pp. 117–130.
7. Grigoriy Blekherman and Zach Teitler, On maximum, typical and generic ranks, arXiv:1402.2371.
8. Nader H. Bshouty, On the direct sum conjecture in the straight line model, J. Complexity 14 (1998), no. 1,
49–62. MR 1617757 (99c:13056)
9. Jaroslaw Buczyński, Adam Ginensky, and J. M. Landsberg, Determinantal equations for secant varieties and
the Eisenbud-Koh-Stillman conjecture, J. Lond. Math. Soc. (2) 88 (2013), no. 1, 1–24. MR 3092255
10. Jaroslaw Buczyński and J. M. Landsberg, Ranks of tensors and a generalization of secant varieties, Linear
Algebra Appl. 438 (2013), no. 2, 668–689. MR 2996361
36
J.M. LANDSBERG AND MATEUSZ MICHALEK
11.
, On the third secant variety, J. Algebraic Combin. 40 (2014), no. 2, 475–502. MR 3239293
12. Peter Bürgisser, Michael Clausen, and M. Amin Shokrollahi, Algebraic complexity theory, Grundlehren
der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 315, SpringerVerlag, Berlin, 1997, With the collaboration of Thomas Lickteig. MR 99c:68002
13. Enrico Carlini, Maria Virginia Catalisano, and Luca Chiantini, Progress on the symmetric Strassen conjecture,
J. Pure Appl. Algebra 219 (2015), no. 8, 3149–3157. MR 3320211
14. P. Comon, Independent Component Analysis, a new concept ?, Signal Processing, Elsevier 36 (1994), no. 3,
287–314, Special issue on Higher-Order Statistics.
15. Don Coppersmith and Shmuel Winograd, Matrix multiplication via arithmetic progressions, J. Symbolic Comput. 9 (1990), no. 3, 251–280. MR 91i:68058
16. Ephraim Feig and Shmuel Winograd, On the direct sum conjecture, Linear Algebra Appl. 63 (1984), 193–219.
MR 766508 (86h:15022)
17. Ismor Fischer, Sums of Like Powers of Multivariate Linear Forms, Math. Mag. 67 (1994), no. 1, 59–61.
MR 1573008
18. Shmuel Friedland, On tensors of border rank l in Cm×n×l , Linear Algebra Appl. 438 (2013), no. 2, 713–737.
MR 2996364
19. Shmuel Friedland and Elizabeth Gross, A proof of the set-theoretic version of the salmon conjecture, J. Algebra
356 (2012), 374–379. MR 2891138
20. Murray Gerstenhaber, On dominance and varieties of commuting matrices, Ann. of Math. (2) 73 (1961),
324–348. MR 0132079 (24 #A1926)
21. Robert M. Guralnick, A note on commuting pairs of matrices, Linear and Multilinear Algebra 31 (1992),
no. 1-4, 71–75. MR 1199042 (94c:15021)
22. Wolfgang Hackbusch, Tensor spaces and numerical tensor calculus, Springer Series in Computational Mathematics, vol. 42, Springer, Heidelberg, 2012. MR 3236394
23. Atanas Iliev and Laurent Manivel, Varieties of reductions for gln , Projective varieties with unexpected properties, Walter de Gruyter GmbH & Co. KG, Berlin, 2005, pp. 287–316. MR MR2202260 (2006j:14056)
24. Atanas Iliev and Kristian Ranestad, Addendum to “K3 surfaces of genus 8 and varieties of sums of powers of
cubic fourfolds” [Trans. Amer. Math. Soc. 353 (2001), no. 4, 1455–1468; mr1806733], C. R. Acad. Bulgare
Sci. 60 (2007), no. 12, 1265–1270. MR MR2391437
25. Joseph Ja’Ja’ and Jean Takche, On the validity of the direct sum conjecture, SIAM J. Comput. 15 (1986),
no. 4, 1004–1020. MR 861366 (88b:68084)
26. Thomas J. Laffey, The minimal dimension of maximal commutative subalgebras of full matrix algebras, Linear
Algebra Appl. 71 (1985), 199–212. MR 813045 (87a:15025)
27. J. M. Landsberg, Tensors: geometry and applications, Graduate Studies in Mathematics, vol. 128, American
Mathematical Society, Providence, RI, 2012. MR 2865915
28.
, New lower bounds for the rank of matrix multiplication, SIAM J. Comput. 43 (2014), no. 1, 144–149.
MR 3162411
29. J. M. Landsberg and Laurent Manivel, Generalizations of Strassen’s equations for secant varieties of Segre
varieties, Comm. Algebra 36 (2008), no. 2, 405–422. MR MR2387532
30. J. M. Landsberg and Zach Teitler, On the ranks and border ranks of symmetric tensors, Found. Comput.
Math. 10 (2010), no. 3, 339–366. MR 2628829 (2011d:14095)
31. J.M. Landsberg and Mateusz Michalek, Compression genericity of tensors, in preparation.
32. François Le Gall, Algebraic complexity theory and matrix multiplication, Proceedings ISSAC, 2014, pp. 23–23.
33. A. Leitner, Limits Under Conjugacy of the Diagonal Subgroup in SL(n,R), ArXiv e-prints (2014).
34. Thomas Lickteig, Typical tensorial rank, Linear Algebra Appl. 69 (1985), 95–120. MR 87f:15017
35. Giorgio Ottaviani, Symplectic bundles on the plane, secant varieties and Lüroth quartics revisited, Vector Bundles and Low Codimensional Subvarieties: State of the Art and Recent Developments (R. Notari G. Casnati,
F. Catanese, ed.), Quaderni di Matematica, vol. 21, Dip. di Mat., II Univ. Napoli, 2007, pp. 315–352.
36. Ran Raz, Tensor-rank and lower bounds for arithmetic formulas, J. ACM 60 (2013), no. 6, Art. 40, 15.
MR 3144910
37. A. Schönhage, Partial and total matrix multiplication, SIAM J. Comput. 10 (1981), no. 3, 434–455.
MR MR623057 (82h:68070)
38. A. Stothers, On the complexity of matrix multiplication, PhD thesis, University of Edinburgh, 2010.
39. V. Strassen, Rank and optimal computation of generic tensors, Linear Algebra Appl. 52/53 (1983), 645–685.
MR 85b:15039
ABELIAN TENSORS
37
40.
, Relative bilinear complexity and matrix multiplication, J. Reine Angew. Math. 375/376 (1987),
406–443. MR MR882307 (88h:11026)
41. Volker Strassen, Vermeidung von Divisionen, J. Reine Angew. Math. 264 (1973), 184–202. MR MR0521168
(58 #25128)
42. D. A. Suprunenko and R. I. Tyshkevich, Perestanovochnye matritsy, second ed., Èditorial URSS, Moscow,
English translation of first edition: Academic Press: New York, 1968, 2003. MR 2118458 (2006b:16045)
43. Virginia Vassilevska Williams, Multiplying matrices faster than Coppersmith-Winograd [extended abstract],
STOC’12—Proceedings of the 2012 ACM Symposium on Theory of Computing, ACM, New York, 2012,
pp. 887–898. MR 2961552
Department of Mathematics, Texas A&M University, Mailstop 3368, College Station, TX 778433368, USA
E-mail address: jml@math.tamu.edu
Freie Universität, Arnimallee 3, 14195 Berlin, Germany
Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warsaw, Poland
E-mail address: wajcha2@poczta.onet.pl